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1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> Green-Schwarz action functional </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/2024/#Item_12" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" 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href="/nlab/show/quantum+field+theory">quantum field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a>,</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/CFT">CFT</a>, <a class="existingWikiWord" href="/nlab/show/perturbation+theory">perturbation theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+QFT">effective background QFT</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/gravity">gravity</a>, <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a>, <a class="existingWikiWord" href="/nlab/show/quantum+gravity">quantum gravity</a></li> </ul> </li> </ul> <h3 id="critical_string_models">Critical string models</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a>, <a class="existingWikiWord" href="/nlab/show/differential+string+structure">differential string structure</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dual+heterotic+string+theory">dual heterotic string theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+fivebrane+structure">differential fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/type+IIA+string+theory">type IIA string theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/type+IIB+string+theory">type IIB string theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/F-theory">F-theory</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+field+theory">string field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/duality+in+string+theory">duality in string theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/T-duality">T-duality</a>, <a class="existingWikiWord" href="/nlab/show/mirror+symmetry">mirror symmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/S-duality">S-duality</a>, <a class="existingWikiWord" href="/nlab/show/electric-magnetic+duality">electric-magnetic duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/U-duality">U-duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open%2Fclosed+string+duality">open/closed string duality</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/AdS%2FCFT+correspondence">AdS/CFT correspondence</a>, <a class="existingWikiWord" href="/nlab/show/holographic+principle">holographic principle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a>, <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%C5%99ava-Witten+theory">Hořava-Witten theory</a></li> </ul> </li> </ul> <h3 id="extended_objects">Extended objects</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/brane">brane</a></p> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/D0-brane">D0-brane</a>, <a class="existingWikiWord" href="/nlab/show/D2-brane">D2-brane</a>, <a class="existingWikiWord" href="/nlab/show/D4-brane">D4-brane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D1-brane">D1-brane</a>, <a class="existingWikiWord" href="/nlab/show/D3-brane">D3-brane</a>, <a class="existingWikiWord" href="/nlab/show/D5-brane">D5-brane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/RR-field">RR-field</a>, <a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/NS-brane">NS-brane</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/string">string</a>, <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spinning+string">spinning string</a>, <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/B2-field">B2-field</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/NS5-brane">NS5-brane</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/B6-field">B6-field</a></li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/M-brane">M-brane</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/C3-field">C3-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ABJM+theory">ABJM theory</a>, <a class="existingWikiWord" href="/nlab/show/BLG+model">BLG model</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/C6-field">C6-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a></p> </li> </ul> </li> </ul> </li> </ul> <h3 id="topological_strings">Topological strings</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a>, <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+M-theory">topological M-theory</a></p> </li> </ul> <h2 id="backgrounds">Backgrounds</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/target+space">target space</a>, <a class="existingWikiWord" href="/nlab/show/background+gauge+field">background gauge field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+smooth+cohomology+in+string+theory">twisted smooth cohomology in string theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/landscape+of+string+theory+vacua">landscape of string theory vacua</a></p> </li> </ul> <h2 id="phenomenology">Phenomenology</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/string+phenomenology">string phenomenology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/moduli+stabilization">moduli stabilization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/G%E2%82%82-MSSM">G₂-MSSM</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/string+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="supergeometry">Super-Geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a></strong> and (<a class="existingWikiWord" href="/nlab/show/synthetic+differential+supergeometry">synthetic</a> ) <strong><a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a></strong></p> <h2 id="background">Background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra">algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/graded+object">graded object</a></p> </li> </ul> <h2 id="introductions">Introductions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+superalgebra">geometry of physics – superalgebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+supergeometry">geometry of physics – supergeometry</a></p> </li> </ul> <h2 id="superalgebra">Superalgebra</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/super+commutative+monoid">super commutative monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+abelian+group">super abelian group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+ring">super ring</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supercommutative+ring">supercommutative ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exterior+ring">exterior ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Clifford+ring">Clifford ring</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+module">super module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+vector+space">super vector space</a>, <a class="existingWikiWord" href="/nlab/show/SVect">SVect</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+algebra">super algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supercommutative+algebra">supercommutative algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exterior+algebra">exterior algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Clifford+algebra">Clifford algebra</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/superdeterminant">superdeterminant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">super Poincare Lie algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+complex+of+super+vector+spaces">chain complex of super vector spaces</a> (<a class="existingWikiWord" href="/nlab/show/model+structure+on+chain+complexes+of+super+vector+spaces">model structure</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+graded-commutative+superalgebra">differential graded-commutative superalgebra</a> (<a class="existingWikiWord" href="/nlab/show/model+structure+on+differential+graded-commutative+superalgebras">model structure</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+L-infinity+algebra">super L-infinity algebra</a></p> </li> </ul> <h2 id="supergeometry">Supergeometry</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/superpoint">superpoint</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Cartesian+space">super Cartesian space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supermanifold">supermanifold</a>, <a class="existingWikiWord" href="/nlab/show/SDiff">SDiff</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/NQ-supermanifold">NQ-supermanifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+vector+bundle">super vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+supermanifold">complex supermanifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+supermanifold">Euclidean supermanifold</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+spacetime">super spacetime</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Minkowski+spacetime">super Minkowski spacetime</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integration+over+supermanifolds">integration over supermanifolds</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Berezin+integral">Berezin integral</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Lie+group">super Lie group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Lie+group">super translation group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Euclidean+group">super Euclidean group</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+%E2%88%9E-groupoid">super ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+formal+smooth+%E2%88%9E-groupoid">super formal smooth ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+line+2-bundle">super line 2-bundle</a></p> </li> </ul> <h2 id="supersymmetry">Supersymmetry</h2> <p><a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/division+algebra+and+supersymmetry">division algebra and supersymmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">super Poincare Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supermultiplet">supermultiplet</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BPS+state">BPS state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/M-theory+super+Lie+algebra">M-theory super Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/type+II+super+Lie+algebra">type II super Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a>, <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+6-algebra">supergravity Lie 6-algebra</a></p> </li> </ul> <h2 id="supersymmetric_field_theory">Supersymmetric field theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/superfield">superfield</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supersymmetric+quantum+mechanics">supersymmetric quantum mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adinkra">adinkra</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Yang-Mills+theory">super Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/gauged+supergravity">gauged supergravity</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/superstring+theory">superstring theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a></p> </li> </ul> </li> </ul> <h2 id="applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/geometric+model+for+elliptic+cohomology">geometric model for elliptic cohomology</a></li> </ul> <div> <p> <a href="/nlab/edit/supergeometry+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="quantum_field_theory">Quantum field theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/FQFT">functorial quantum field theory</a></strong></p> <h2 id="contents">Contents</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+category">cobordism category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+cobordism">extended cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bordism+categories+following+Stolz-Teichner">Riemannian bordism category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+tangle+hypothesis">generalized tangle hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/On+the+Classification+of+Topological+Field+Theories">classification of TQFTs</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functorial+field+theory">functorial field theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+functorial+field+theory">unitary functorial field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+functorial+field+theory">extended functorial field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+field+theory">CFT</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/vertex+operator+algebra">vertex operator algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Reshetikhin-Turaev+model">Reshetikhin-Turaev model</a> / <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/HQFT">HQFT</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a>, <a class="existingWikiWord" href="/nlab/show/Gromov-Witten+theory">Gromov-Witten theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p>FQFT and <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory">(1,1)-dimensional Euclidean field theories and K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+models+for+tmf">geometric models for tmf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holographic+principle+of+higher+category+theory">holographic principle of higher category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/holographic+principle">holographic principle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/AdS%2FCFT+correspondence">AdS/CFT correspondence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantization+via+the+A-model">quantization via the A-model</a></p> </li> </ul> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#Idea'>Idea</a></li> <ul> <li><a href='#broad_idea'>Broad idea</a></li> <li><a href='#more_details'>More details</a></li> </ul> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#Supercoordinates'>Supercoordinates</a></li> <li><a href='#kinetic_term'>Kinetic term</a></li> <li><a href='#DefinitionWZWTerm'>WZW term</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#KappaSymmetry'>Siegel- or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>-symmetry</a></li> <li><a href='#BraneScan'>Dimensions – the brane scan</a></li> <li><a href='#the_brane_molecule'>The Brane molecule</a></li> <li><a href='#OnCurvedSpacetime'>On curved spacetime and supergravity equations of motion</a></li> <ul> <li><a href='#MembraneIn11dSuGraBackground'>Membrane in 11d SuGra background</a></li> <li><a href='#heterotic_string'>Heterotic string</a></li> <li><a href='#type_ii_string'>Type II string</a></li> </ul> <li><a href='#ConservedCurrents'>Conserved currents</a></li> <li><a href='#AsPartOfTheAdSCFTCorrespodence'>As part of the AdS-CFT correspondence</a></li> <li><a href='#quantization'>Quantization</a></li> </ul> <li><a href='#related_entries'>Related entries</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#SuperStringAsAGSSigmaModel'>Super-string as a GS-sigma model</a></li> <li><a href='#SuperMembraneAsSigmaModelReferences'>Super-membrane/M2-brane as a sigma model</a></li> <li><a href='#ReferencesForDBranes'>D-branes as GS-sigma models</a></li> <li><a href='#dualities'>Dualities</a></li> <li><a href='#ReferencesWZWTerm'>WZW terms, super Lie algebra cohomology and the brane scan</a></li> <li><a href='#ReferencesAdSBackgrounds'>Anti de Sitter backgrounds</a></li> <ul> <li><a href='#ReferencesAdS5Background'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>AdS</mi> <mn>5</mn></msub></mrow><annotation encoding="application/x-tex">AdS_5</annotation></semantics></math></a></li> <li><a href='#ReferencesAdS4Background'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>AdS</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">AdS_4</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>AdS</mi> <mn>7</mn></msub></mrow><annotation encoding="application/x-tex">AdS_7</annotation></semantics></math></a></li> </ul> <li><a href='#selfdual_strings_in_6d'>Self-dual strings in 6d</a></li> <li><a href='#ReferencesSupergravityBackgroundEquationsOfMotion'>General curved backgrounds and Supergravity background equations of motion</a></li> <li><a href='#ReferencesMicroscopicAdSCFTViapBraneSigmaModesl'>Microscopic AdS/CFT via <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-brane sigma-models</a></li> <li><a href='#ReferencesConservedCurrentAlgebra'>Conserved current algebra</a></li> <li><a href='#symmetry'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>-Symmetry</a></li> <li><a href='#open_branes_ending_on_other_branes'>Open branes ending on other branes</a></li> </ul> </ul> </div> <h2 id="Idea">Idea</h2> <h3 id="broad_idea">Broad idea</h3> <p>The <em>Green-Schwarz action functional</em> is an <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> for a <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> that describes the propagation of a fundamental super <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/brane">brane</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> in a <a class="existingWikiWord" href="/nlab/show/super+spacetime">super spacetime</a> <a class="existingWikiWord" href="/nlab/show/supermanifold">supermanifold</a>.</p> <ul> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">p = 0</annotation></semantics></math> this is the <strong>Green-Schwarz <a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a></strong>.</p> </li> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">p = 1</annotation></semantics></math> this is the <strong>Green-Schwarz <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a></strong> (<a href="#GreenSchwarz84">Green-Schwarz 84</a>)</p> </li> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">p =2</annotation></semantics></math> this is the <strong>Green-Schwarz <a class="existingWikiWord" href="/nlab/show/supermembrane">supermembrane</a></strong> (<a href="#BergshoeffSezginTownsend87">Bergshoeff-Sezgin-Townsend 87</a>)</p> </li> </ul> <p>The Green-Schwarz model of the superstring is in contrast to the <em><a class="existingWikiWord" href="/nlab/show/NSR-string">NSR-string</a></em> model (the original <a class="existingWikiWord" href="/nlab/show/spinning+string">spinning string</a>), which has manifest <a class="existingWikiWord" href="/nlab/show/worldsheet">worldsheet</a> <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a> but no manifest spacetime supersymmetry. It is a non-trivial theorem that the spectrum of the <a class="existingWikiWord" href="/nlab/show/NSR-string">NSR-string</a> enjoys spacetime supersymmetry (after <a class="existingWikiWord" href="/nlab/show/GSO+projection">GSO projection</a>) and may hence be identified with <a class="existingWikiWord" href="/nlab/show/perturbation+theory">perturbative</a> excitations of a <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> background. The construction of the Green-Schwarz functional was motivated by the desire to find an equivalent alternative formulation in which spacetime supersymmetry is manifest (see e.g. <a href="#Schwarz16">Schwarz 16, slides 24-25</a>).</p> <div> <p><strong>manifest <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a> for <a class="existingWikiWord" href="/nlab/show/brane">brane</a> <a class="existingWikiWord" href="/nlab/show/sigma-models">sigma-models</a></strong>:</p> <table><thead><tr><th>manifest <a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a> supersymmetry</th><th>manifest target+worldvolume supersymmetry</th><th>manifest <a class="existingWikiWord" href="/nlab/show/target+space">target space</a> supersymmetry</th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/NSR+action+functional">NSR action functional</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/superembedding+approach">superembedding approach</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+action+functional">Green-Schwarz action functional</a></td></tr> </tbody></table> <center> <img src="https://ncatlab.org/nlab/files/pBraneEmbedding.jpg" width="800" /> </center> <blockquote> <p>graphics grabbed from <a class="existingWikiWord" href="/schreiber/show/Super-exceptional+embedding+construction+of+the+M5-brane">FSS19c</a></p> </blockquote> </div> <p>For more discussion see also at <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+fundamental+super+p-branes">geometry of physics – fundamental super p-branes</a></em>.</p> <h3 id="more_details">More details</h3> <p><a class="existingWikiWord" href="/nlab/show/perturbation+theory">Perturbative</a><a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> on geometric backgrounds is defined by the <a class="existingWikiWord" href="/nlab/show/Neveu-Schwarz-Ramond+model">Neveu-Schwarz-Ramond model</a>, namely by <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> <a class="existingWikiWord" href="/nlab/show/2d+super+conformal+field+theories">2d super conformal field theories</a> (of <a class="existingWikiWord" href="/nlab/show/central+charge">central charge</a> 15) on <a class="existingWikiWord" href="/nlab/show/worldsheets">worldsheets</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> that are <a class="existingWikiWord" href="/nlab/show/super+Riemann+surfaces">super Riemann surfaces</a>, with <a class="existingWikiWord" href="/nlab/show/target+spaces">target spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> that are ordinary (i.e. “bosonic”) <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <a class="existingWikiWord" href="/nlab/show/manifolds">manifolds</a>.</p> <p>These <a class="existingWikiWord" href="/nlab/show/worldsheet">worldsheet</a> <a class="existingWikiWord" href="/nlab/show/field+theories">field theories</a> are induced from <em><a class="existingWikiWord" href="/nlab/show/action+functionals">action functionals</a></em>, namely from variants of the standard <a class="existingWikiWord" href="/nlab/show/energy+functional">energy functional</a> (<a class="existingWikiWord" href="/nlab/show/Polyakov+action">Polyakov action</a>) on the <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\Sigma,X]</annotation></semantics></math> of smooth functions</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Σ</mi><mo>⟶</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> \phi \;\colon\; \Sigma \longrightarrow X </annotation></semantics></math></div> <p>from the <a class="existingWikiWord" href="/nlab/show/worldsheet">worldsheet</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> to target <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>The central theorem of perturbative superstring theory (the no ghost theorem with <a class="existingWikiWord" href="/nlab/show/GSO+projection">GSO projection</a>) says that the excitation spectrum of such a <a class="existingWikiWord" href="/nlab/show/2d+SCFT">2d SCFT</a> are the quanta of the <a class="existingWikiWord" href="/nlab/show/perturbation+theory">perturbations</a> of a higher dimensional <a class="existingWikiWord" href="/nlab/show/effective+field+theory">effective</a> <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> <a class="existingWikiWord" href="/nlab/show/field+theory">field theory</a> on target spacetime, hence transforms under <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a> on target spacetime.</p> <p>This is the fundamental prediction of the assumption of fundamental strings:</p> <ol> <li> <p>assuming that the <a class="existingWikiWord" href="/nlab/show/fundamental+particles">fundamental particles</a> that run in <a class="existingWikiWord" href="/nlab/show/Feynman+diagrams">Feynman diagrams</a> are fundamentally (at high energy) the ground state modes of a fundamental <a class="existingWikiWord" href="/nlab/show/string">string</a>,</p> </li> <li> <p>demanding that there are <a class="existingWikiWord" href="/nlab/show/fermion">fermionic</a> particles among these,</p> </li> </ol> <p>implies</p> <ol> <li> <p>that the string must be the <a class="existingWikiWord" href="/nlab/show/spinning+string">spinning string</a> (have fermions in its <a class="existingWikiWord" href="/nlab/show/worldsheet">worldsheet</a> theory), which in turn implies…</p> </li> <li> <p>that it is the <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a> (worldsheet <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a> mixes the worldsheet bosons and fermions), precisely: the <a class="existingWikiWord" href="/nlab/show/Neveu-Schwarz-Ramond+superstring">Neveu-Schwarz-Ramond superstring</a>, which then in addition implies…</p> </li> <li> <p>that its <a class="existingWikiWord" href="/nlab/show/target+space">target space</a> <a class="existingWikiWord" href="/nlab/show/effective+field+theory">effective field theory</a> is a <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> theory, hence that also the effective target space fields exhibit <em>local</em> <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a> (i.e. “high energy supersymmetry”, different from “low energy supersymmetry” that the <a class="existingWikiWord" href="/nlab/show/LHC">LHC</a> was looking for).</p> </li> </ol> <table><thead><tr><th>main theorem of <a class="existingWikiWord" href="/nlab/show/perturbative+string+theory">perturbative super-string theory</a></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><munder><mrow><mtext>fermions</mtext><mspace width="thickmathspace"></mspace><mo>+</mo><mspace width="thickmathspace"></mspace><mtext>strings</mtext></mrow><mo>⏟</mo></munder><mtext>spinning string</mtext></munder><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mtext>superstring</mtext><mspace width="thickmathspace"></mspace><mo>⇒</mo><mspace width="thickmathspace"></mspace><mtext>supergravity</mtext></mrow><annotation encoding="application/x-tex"> \underset{\text{spinning string}}{\underbrace{\text{fermions} \;+\; \text{strings}}} \;=\; \text{superstring} \;\Rightarrow\; \text{supergravity} </annotation></semantics></math></td></tr> </tbody></table> <p>The first step in this implication (identifying the <a class="existingWikiWord" href="/nlab/show/spinning+string">spinning string</a> as the <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a>) is fairly straightforward (in fact this is how the concept of <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a> was discovered in “the west”, in the first place), but the second step (that the superstring excitations necessarily are quanta of a <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> theory) appears as a miracle from the point of view of the <a class="existingWikiWord" href="/nlab/show/Neveu-Schwarz-Ramond+superstring">Neveu-Schwarz-Ramond superstring</a>. It comes out this way by non-trivial computation, but is not manifest in the theory.</p> <p>In order to improve on this situation, <a class="existingWikiWord" href="/nlab/show/Michael+Green">Michael Green</a> and <a class="existingWikiWord" href="/nlab/show/John+Schwarz">John Schwarz</a> searched for and found (<a href="#GreenSchwarz81">Green-Schwarz 81</a>, <a href="#GreenSchwarz82">Green-Schwarz 82</a> <a href="#GreenSchwarz84">Green-Schwarz 84</a>, for the history see <a href="#Schwarz16">Schwarz 16, slides 24-25</a>) a suitably equivalent string <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> that would manifestly exhibit <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a>. Acordingly, this is now called the <em><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+action+functional">Green-Schwarz action functional</a></em>.</p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> for <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a></th><th>manifest <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Neveu-Ramond-Schwarz+super-string">Neveu-Ramond-Schwarz super-string</a></td><td style="text-align: left;">on <a class="existingWikiWord" href="/nlab/show/worldsheet">worldsheet</a></td></tr> <tr><td style="text-align: left;">Green-Schwarz super-string</td><td style="text-align: left;">on <a class="existingWikiWord" href="/nlab/show/target+space">target</a> <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a></td></tr> </tbody></table> <p>The basic idea is to pass to the evident <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometric</a> analogue of the bosonic string action:</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/closed+manifold">closed manifold</a> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> 2 – representing the abstract <em><a class="existingWikiWord" href="/nlab/show/worldsheet">worldsheet</a></em> of a string. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,g)</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+manifold">pseudo-Riemannian manifold</a> – representing a purely <a class="existingWikiWord" href="/nlab/show/gravity">gravitational</a> <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> background. Then the <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> governing the <a class="existingWikiWord" href="/nlab/show/bosonic+string">bosonic string</a> propagating in this spacetime is the functional</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mi>ℏ</mi></mfrac></mstyle><msub><mi>S</mi> <mi>bos</mi></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo>⟶</mo><mi>ℝ</mi><msub><mo stretchy="false">/</mo> <mi>ℏ</mi></msub><mi>ℤ</mi></mrow><annotation encoding="application/x-tex"> \exp(\tfrac{i}{\hbar} S_{bos}) \;\colon\; [\Sigma,X] \longrightarrow \mathbb{R}/_{\hbar}\mathbb{Z} </annotation></semantics></math></div> <p>on the <a class="existingWikiWord" href="/nlab/show/smooth+space">smooth</a> <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\Sigma,X]</annotation></semantics></math> (of <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\Sigma \to X</annotation></semantics></math>), that simply assigns the proper <a class="existingWikiWord" href="/nlab/show/relativity">relativistic</a> <a class="existingWikiWord" href="/nlab/show/volume">volume</a> of the image of the <a class="existingWikiWord" href="/nlab/show/worldsheet">worldsheet</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> in <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Σ</mi><mover><mo>⟶</mo><mi>ϕ</mi></mover><mi>X</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>↦</mo><mspace width="thickmathspace"></mspace><msub><mi>S</mi> <mi>kin</mi></msub><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo>≔</mo><msub><mo>∫</mo> <mi>Σ</mi></msub><msub><mi>vol</mi> <mrow><msup><mi>ϕ</mi> <mo>*</mo></msup><mi>g</mi></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\Sigma\overset{\phi}{\longrightarrow} X) \;\mapsto\; S_{kin}(\phi) \coloneqq \int_\Sigma vol_{\phi^\ast g} \,. </annotation></semantics></math></div> <p>(This is the <em><a class="existingWikiWord" href="/nlab/show/Nambu-Goto+action">Nambu-Goto action</a></em>. It is classically equivalent to the <a class="existingWikiWord" href="/nlab/show/Polyakov+action">Polyakov action</a> which is the genuine starting point for the quantum <a class="existingWikiWord" href="/nlab/show/Neveu-Ramond-Schwarz+super-string">Neveu-Ramond-Schwarz super-string</a>. However, since, as we discuss below, the Green-Schwarz action naturally generalizes to that of other <a class="existingWikiWord" href="/nlab/show/p-branes">p-branes</a> it is more natural to consider the Nambu-Goto form of the action here.)</p> <p>When here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,g)</annotation></semantics></math> is generalized to a <a class="existingWikiWord" href="/nlab/show/superspacetime">superspacetime</a> <a class="existingWikiWord" href="/nlab/show/supermanifold">supermanifold</a> with <a class="existingWikiWord" href="/nlab/show/orthogonal+structure">orthogonal structure</a> encoded by a <a class="existingWikiWord" href="/nlab/show/super-vielbein">super-vielbein</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math>, then the same form of the action functional still makes sense and produces a functional on the <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometric</a> <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\Sigma,X]</annotation></semantics></math>. Moreover, by construction this action functional now is invariant under the <a class="existingWikiWord" href="/nlab/show/superisometry+group">superisometry group</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,g)</annotation></semantics></math>, hence under global spacetime <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mtext>symmetry of worldsheet theory</mtext></mtd></mtr> <mtr><mtd><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>Σ</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>ϕ</mi></mpadded></msup><mo>↙</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><mi>ϕ</mi><mo>′</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><munder><mo>⟶</mo><mo>≃</mo></munder></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow></mtd></mtr> <mtr><mtd><mtext>super-isometry of target spacetime</mtext></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \text{symmetry of worldsheet theory} \\ \array{ && \Sigma \\ & {}^{\mathllap{\phi}}\swarrow && \searrow^{\mathrlap{\phi'}} \\ X &&\underset{\simeq}{\longrightarrow}&& X } \\ \text{super-isometry of target spacetime} } </annotation></semantics></math></div> <p>However, <a class="existingWikiWord" href="/nlab/show/Michael+Green">Green</a> and <a class="existingWikiWord" href="/nlab/show/John+Schwarz">Schwarz</a> noticed that this <a class="existingWikiWord" href="/nlab/show/kinetic+action">kinetic action</a> functional <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>↦</mo><msub><mo>∫</mo> <mi>Σ</mi></msub><msub><mi>vol</mi> <mrow><msup><mi>ϕ</mi> <mo>*</mo></msup><mi>e</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\phi \mapsto \int_\Sigma vol_{\phi^\ast e}</annotation></semantics></math> does <em>not</em> quite yield dynamics that is equivalent to that of the <a class="existingWikiWord" href="/nlab/show/Neveu-Schwarz-Ramond+super-string">Neveu-Schwarz-Ramond super-string</a>: when the <a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a> hold (“on <a class="existingWikiWord" href="/nlab/show/shell">shell</a>”) it has more fermionic degrees of freedom than present in the <a class="existingWikiWord" href="/nlab/show/Neveu-Ramond-Schwarz+super-string">Neveu-Ramond-Schwarz super-string</a>. The key insight of Green and Schwarz was that one may add an extra summand <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>WZW</mi></msub></mrow><annotation encoding="application/x-tex">S_{WZW}</annotation></semantics></math> (whose notation we explain in a moment) to the plain super-<a class="existingWikiWord" href="/nlab/show/Nambu-Goto+action">Nambu-Goto action</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>kin</mi></msub></mrow><annotation encoding="application/x-tex">S_{kin}</annotation></semantics></math>, such that the resulting <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> enjoys a further 1-parameter <a class="existingWikiWord" href="/nlab/show/symmetry">symmetry</a>, called <em><a class="existingWikiWord" href="/nlab/show/kappa-symmetry">kappa-symmetry</a></em>. This is the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+action+functional">Green-Schwarz action functional</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>GS</mi></msub><mo>=</mo><msub><mi>S</mi> <mi>kin</mi></msub><mo>+</mo><msub><mi>S</mi> <mi>WZW</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> S_{GS} = S_{kin} + S_{WZW} \,. </annotation></semantics></math></div> <p>Moreover, they showed that restricting the <a class="existingWikiWord" href="/nlab/show/dynamics">dynamics</a> of the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+superstring">Green-Schwarz superstring</a> to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>-symmetric states, then it does become equivalent, classically to that of the <a class="existingWikiWord" href="/nlab/show/Neveu-Ramond-Schwarz+super-string">Neveu-Ramond-Schwarz super-string</a>.</p> <p>Finally they showed that when <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixing</a> the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+action+functional">Green-Schwarz action functional</a> to <a class="existingWikiWord" href="/nlab/show/light-cone+gauge">light-cone gauge</a> (which is possible whenever <a class="existingWikiWord" href="/nlab/show/target+spacetime">target spacetime</a> admits two <a class="existingWikiWord" href="/nlab/show/lightlike">lightlike</a> <a class="existingWikiWord" href="/nlab/show/Killing+vector">Killing vector</a>) then the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+string">Green-Schwarz string</a> may be <a class="existingWikiWord" href="/nlab/show/quantization">quantized</a> by a standard procedure and the resulting <a class="existingWikiWord" href="/nlab/show/quantum+system">quantum</a> dynamics is equivalent to that of the <a class="existingWikiWord" href="/nlab/show/Neveu-Schwarz-Ramond+super-string">Neveu-Schwarz-Ramond super-string</a>. This provides the desired conceptual <a class="existingWikiWord" href="/nlab/show/proof">proof</a> for the observed local <a class="existingWikiWord" href="/nlab/show/target+spacetime">target spacetime</a> <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a> of super-string <a class="existingWikiWord" href="/nlab/show/effective+field+theory">effective field theory</a>, at least for backgrounds that admit two lightlike Killing vectors. (The <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> of the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+superstring">Green-Schwarz superstring</a> away from <a class="existingWikiWord" href="/nlab/show/light+cone+gauge">light cone gauge</a> remains an open problem.)</p> <p>While Green-Schwarz’s extra <a class="existingWikiWord" href="/nlab/show/kappa-symmetry">kappa-symmetry</a> term <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>WZW</mi></msub></mrow><annotation encoding="application/x-tex">S_{WZW}</annotation></semantics></math> this serves a clear purpose as a means to an end, originally its geometric meaning was mysterious. However, in (<a href="#HenneauxMezincescu85">Henneaux-Mezincescu 85</a>) it was observed (expanded on in (<a href="#Rabin87">Rabin 87</a>, <a href="#AzcarragaTownsend89">Azcarraga-Townsend 89</a>, <a href="#AzcarragaIzquierdo95">Azcarraga-Izquierdo 95,chapter 8</a>)), that the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+action+functional">Green-Schwarz action functional</a> describing the <a class="existingWikiWord" href="/nlab/show/super-string">super-string</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">d+1</annotation></semantics></math>-dimensions does have a neat geometrical interpretation: it is simply the (<a class="existingWikiWord" href="/nlab/show/parameterized+WZW+model">parameterized</a>) <em><a class="existingWikiWord" href="/nlab/show/Wess-Zumino-Witten+model">Wess-Zumino-Witten model</a></em> for</p> <ol> <li> <p><a class="existingWikiWord" href="/nlab/show/target+space">target space</a> being locally <a class="existingWikiWord" href="/nlab/show/super+Minkowski+spacetime">super Minkowski spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mi>N</mi></mstyle></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d-1,1|\mathbf{N}}</annotation></semantics></math> regarded as the <a class="existingWikiWord" href="/nlab/show/coset">coset</a> <a class="existingWikiWord" href="/nlab/show/supergroup">supergroup</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mi>N</mi></mstyle></mrow></msup><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>Iso</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mi>N</mi></mstyle></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>Spin</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbb{R}^{d-1,1\vert \mathbf{N}} \;\simeq\; Iso(\mathbb{R}^{d-1,1\vert \mathbf{N}}) / Spin(d-1,1) </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>N</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{N}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/real+spin+representation">real spin representation</a> (the “number of supersymmetries”), <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Iso</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mi>N</mi></mstyle></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Iso(\mathbb{R}^{d-1,1\vert \mathbf{N}})</annotation></semantics></math> the corresponding <a class="existingWikiWord" href="/nlab/show/super+Poincar%C3%A9+group">super Poincaré group</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(d-1,1)</annotation></semantics></math> its <a class="existingWikiWord" href="/nlab/show/Lorentzian+manifold">Lorentz-signature</a> <a class="existingWikiWord" href="/nlab/show/spin+group">Spin</a> <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/WZW-term">WZW-term</a> being a local potential for the unique (up to rescaling, if it exists) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(d-1,1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/invariant">invariant</a> <a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">super Lie algebra 3-cocycle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mrow><mi>F</mi><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\mu_{F1}</annotation></semantics></math> on the <a class="existingWikiWord" href="/nlab/show/super+Poincar%C3%A9+Lie+algebra">super Poincaré Lie algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔦𝔰𝔬</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mi>N</mi></mstyle></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{iso}(\mathbb{R}^{d-1,1\vert \mathbf{N}})</annotation></semantics></math>, with components locally given by the Gamma-matrices of the given <a class="existingWikiWord" href="/nlab/show/Clifford+algebra">Clifford algebra</a> representation; in terms of the <a class="existingWikiWord" href="/nlab/show/super+vielbein">super vielbein</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>e</mi> <mi>a</mi></msup><mo>,</mo><msup><mi>ψ</mi> <mi>α</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(e^a, \psi^\alpha)</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mrow><mi>F</mi><mn>1</mn></mrow></msub><mo>=</mo><mover><mi>ψ</mi><mo>¯</mo></mover><mo>∧</mo><msub><mi>Γ</mi> <mi>A</mi></msub><mi>ψ</mi><mo>∧</mo><msub><mi>e</mi> <mi>a</mi></msub></mrow><annotation encoding="application/x-tex"> \mu_{F1} = \overline{\psi} \wedge \Gamma_A \psi \wedge e_a </annotation></semantics></math></div> <p>and so in components the bi-fermionic component of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mrow><mi>F</mi><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\mu_{F1}</annotation></semantics></math> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>μ</mi> <mrow><mi>F</mi><mn>1</mn></mrow></msub><msub><mo stretchy="false">)</mo> <mrow><mi>a</mi><mi>α</mi><mi>β</mi></mrow></msub><mo>=</mo><msub><mi>Γ</mi> <mrow><mi>a</mi><mi>α</mi><mi>β</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> (\mu_{F1})_{a \alpha \beta} = \Gamma_{a \alpha \beta} </annotation></semantics></math></div> <p>and all other components vanish.</p> </li> </ol> <p>More in detail, just as ordinary <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d-1,1}</annotation></semantics></math> may be identified with the <a class="existingWikiWord" href="/nlab/show/translation+group">translation group</a> along itself, with canonical <a class="existingWikiWord" href="/nlab/show/linear+basis">linear basis</a> of <a class="existingWikiWord" href="/nlab/show/left+invariant+1-forms">left invariant 1-forms</a> given by the canonical <a class="existingWikiWord" href="/nlab/show/vielbein">vielbein</a> field</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>e</mi> <mi>a</mi></msup><mo>≔</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mi>a</mi></msup><msubsup><mo stretchy="false">}</mo> <mrow><mi>a</mi><mo>=</mo><mn>0</mn></mrow> <mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msubsup><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \{e^a \coloneqq \mathbf{d}x^a\}_{a = 0}^{d-1} \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>x</mi> <mi>a</mi></msup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{x^a\}</annotation></semantics></math> are the canonical <a class="existingWikiWord" href="/nlab/show/coordinates">coordinates</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d-1,1}</annotation></semantics></math>, so <a class="existingWikiWord" href="/nlab/show/super+Minkowski+spacetime">super Minkowski spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mi>N</mi></mstyle></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d-1,1\vert \mathbf{N}}</annotation></semantics></math> for some <a class="existingWikiWord" href="/nlab/show/real+spin+representation">real spin representation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>N</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{N}</annotation></semantics></math> is characterized as the <a class="existingWikiWord" href="/nlab/show/supergroup">supergroup</a> whose <a class="existingWikiWord" href="/nlab/show/left+invariant+1-forms">left invariant 1-forms</a> constitute the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo>×</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}</annotation></semantics></math>-bigraded differential with generators the <a class="existingWikiWord" href="/nlab/show/super-vielbein">super-vielbein</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><munder><mrow><msup><mi>e</mi> <mi>a</mi></msup></mrow><mo>⏟</mo></munder><mrow><mi>deg</mi><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mi>even</mi><mo stretchy="false">)</mo></mrow></munder><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mi>a</mi></msup><mo>+</mo><mover><mi>θ</mi><mo>¯</mo></mover><msup><mi>Γ</mi> <mi>a</mi></msup><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>θ</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><munder><munder><mrow><msup><mi>ψ</mi> <mi>α</mi></msup></mrow><mo>⏟</mo></munder><mrow><mi>deg</mi><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mi>odd</mi><mo stretchy="false">)</mo></mrow></munder><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>θ</mi> <mi>α</mi></msup><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \underset{deg = (1,even)}{\underbrace{e^a}} \;\coloneqq\; \mathbf{d}x^a + \overline{\theta}\Gamma^a \mathbf{d} \theta \;\;\;\,,\;\;\;\;\;\;\;\;\;\; \underset{deg = (1,odd)}{\underbrace{\psi^\alpha}} \;\coloneqq\; \mathbf{d}\theta^\alpha \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>x</mi> <mi>a</mi></msup><mo>,</mo><msup><mi>θ</mi> <mi>α</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x^a, \theta^\alpha)</annotation></semantics></math> are the canonical coordinates on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mi>N</mi></mstyle></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d-1,1\vert \mathbf{N}}</annotation></semantics></math>, with the odd-graded elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>θ</mi> <mi>α</mi></msup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\theta^\alpha\}</annotation></semantics></math> spanning the given real <a class="existingWikiWord" href="/nlab/show/spin+representation">Spin(d-1,1)-representation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>N</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{N}</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/Clifford+algebra">Clifford algebra</a> generators <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>Γ</mi> <mi>a</mi></msup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\Gamma^a\}</annotation></semantics></math>.</p> <p>Now while ordinary <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d-1,1}</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>, reflected by the fact that its <a class="existingWikiWord" href="/nlab/show/left-invariant+1-forms">left-invariant 1-forms</a> are all closed</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>e</mi> <mi>a</mi></msup><mo>=</mo><mn>0</mn><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>on</mi><mspace width="thickmathspace"></mspace><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathbf{d}e^a = 0 \;\;\;\;\;\; on \; \mathbb{R}^{d-1,1} \,, </annotation></semantics></math></div> <p>the key phenomenon of <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a> (that two <a class="existingWikiWord" href="/nlab/show/fermions">fermions</a> pair to a <a class="existingWikiWord" href="/nlab/show/bosons">bosons</a>) means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mi>N</mi></mstyle></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d-1,1\vert \mathbf{N}}</annotation></semantics></math> is slightly non-abelian, reflected by the fact that the <a class="existingWikiWord" href="/nlab/show/super-vielbein">super-vielbein</a> is not closed</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>e</mi> <mi>a</mi></msup><mo>=</mo><mover><mi>ψ</mi><mo>¯</mo></mover><mo>∧</mo><msup><mi>Γ</mi> <mi>a</mi></msup><mi>ψ</mi><mspace width="thickmathspace"></mspace><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>ψ</mi> <mi>α</mi></msup><mo>=</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{d} e^a = \overline{\psi} \wedge \Gamma^a \psi \;\,,\;\;\;\;\;\; \mathbf{d} \psi^\alpha = 0 \,. </annotation></semantics></math></div> <p>This elementary effect is the source of all the rich structure seen in the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+super-string">Green-Schwarz super-string</a> and generally in all <a class="existingWikiWord" href="/nlab/show/super+p-brane">super p-brane</a> theory. (The above differential is equivalently that in the <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a> of <a class="existingWikiWord" href="/nlab/show/super+Minkowski+spacetime">super Minkowski spacetime</a>, hence its <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> is the super-<a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a> of <a class="existingWikiWord" href="/nlab/show/super+Minkowski+spacetime">super Minkowski spacetime</a>. In parts of the <a class="existingWikiWord" href="/nlab/show/physics">physics</a> literature this is referred to a “<a class="existingWikiWord" href="/nlab/show/tau+cohomology">tau cohomology</a>”.)</p> <p>In particular, for special combinations of <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math> and number of <a class="existingWikiWord" href="/nlab/show/supersymmetries">supersymmetries</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>N</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{N}</annotation></semantics></math> (i.e. <a class="existingWikiWord" href="/nlab/show/real+spin+representation">real spin representation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math>) then the 3-form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mrow><mi>F</mi><mn>1</mn></mrow></msub><mo>=</mo><mover><mi>ψ</mi><mo>¯</mo></mover><mo>∧</mo><msub><mi>Γ</mi> <mi>a</mi></msub><mi>ψ</mi><mo>∧</mo><msup><mi>e</mi> <mi>a</mi></msup></mrow><annotation encoding="application/x-tex"> \mu_{F1} = \overline{\psi} \wedge \Gamma_a \psi \wedge e^a </annotation></semantics></math></div> <p>is a non-trivial <a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super</a> <a class="existingWikiWord" href="/nlab/show/Lie+algebra+cocycle">Lie algebra cocycle</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mi>N</mi></mstyle></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d-1,1\vert \mathbf{N}}</annotation></semantics></math>, in that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><msub><mi>μ</mi> <mrow><mi>F</mi><mn>1</mn></mrow></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\mathbf{d}\mu_{F1} = 0</annotation></semantics></math> and so that there is no <a class="existingWikiWord" href="/nlab/show/left+invariant+differential+form">left invariant differential form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>b</mi><mo>=</mo><msub><mi>μ</mi> <mrow><mi>F</mi><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{d}b = \mu_{F1}</annotation></semantics></math> (beware here the left-invariance condition: there are of course non-left-invariant potentials for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mrow><mi>F</mi><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\mu_{F1}</annotation></semantics></math>, and in fact these are exactly the possible <a class="existingWikiWord" href="/nlab/show/Lagrangian+densities">Lagrangian densities</a> for the WZW action functional <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>WZW</mi></msub></mrow><annotation encoding="application/x-tex">S_{WZW}</annotation></semantics></math>).</p> <p>This happens notably for</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><mn>10</mn></mrow><annotation encoding="application/x-tex">d = 10</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>N</mi></mstyle><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mstyle mathvariant="bold"><mn>16</mn></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{N} = (1,0) = \mathbf{16}</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/heterotic+string">heterotic string</a>)</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><mn>10</mn></mrow><annotation encoding="application/x-tex">d = 10</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>N</mi></mstyle><mo>=</mo><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mstyle mathvariant="bold"><mn>16</mn></mstyle><mo>+</mo><mstyle mathvariant="bold"><mn>16</mn></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{N} = (2,0) = \mathbf{16} + \mathbf{16}</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/type+IIB+superstring">type IIB superstring</a>)</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><mn>10</mn></mrow><annotation encoding="application/x-tex">d = 10</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>N</mi></mstyle><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mstyle mathvariant="bold"><mn>16</mn></mstyle><mo>+</mo><msup><mstyle mathvariant="bold"><mn>16</mn></mstyle> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\mathbf{N} = (1,1) = \mathbf{16} + \mathbf{16}^\ast</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/type+IIA+superstring">type IIA superstring</a>).</p> </li> </ol> <p>(It also happens in some lower dimensions, where however the corresponding <a class="existingWikiWord" href="/nlab/show/Neveu-Schwarz-Ramond+string">Neveu-Schwarz-Ramond string</a> develops a <a class="existingWikiWord" href="/nlab/show/conformal+anomaly">conformal anomaly</a> after <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> (“non-critical strings”). This classification of cocycles is part of what has come to be known as the <em><a class="existingWikiWord" href="/nlab/show/brane+scan">brane scan</a></em> in superstring theory, see below.)</p> <p>In this equivalent formulation, the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+action+functional">Green-Schwarz action functional</a> for the superstring has the following simple form:</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,e)</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/superspacetime">superspacetime</a>, hence a <a class="existingWikiWord" href="/nlab/show/supermanifold">supermanifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> equipped with a <a class="existingWikiWord" href="/nlab/show/super-vielbein">super-vielbein</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math> (super-<a class="existingWikiWord" href="/nlab/show/orthogonal+structure">orthogonal structure</a>) which is locally modeled on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mi>N</mi></mstyle></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d-1,1\vert \mathbf{N}}</annotation></semantics></math> (technically: a <a class="existingWikiWord" href="/nlab/show/torsion+of+a+G-structure">torsion</a>-free <a class="existingWikiWord" href="/nlab/show/super-Cartan+geometry">super-Cartan geometry</a> modeled on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>↪</mo><mi>Iso</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mi>N</mi></mstyle></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(d-1,1) \hookrightarrow Iso(\mathbb{R}^{d-1,1\vert \mathbf{N}})</annotation></semantics></math>). Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>μ</mi> <mrow><mi>F</mi><mn>1</mn></mrow> <mi>X</mi></msubsup><mo>∈</mo><msup><mi>Ω</mi> <mn>3</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mu_{F1}^X \in \Omega^3(X)</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/super+differential+form">super differential form</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> which is the induced <a class="existingWikiWord" href="/nlab/show/definite+globalization+of+WZW+term">definite globalization</a> of the cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mrow><mi>F</mi><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\mu_{F1}</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">U \subset X</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/contractible+topological+space">contractible</a> <a class="existingWikiWord" href="/nlab/show/subspace">subspace</a>, then the restriction of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>μ</mi> <mrow><mi>F</mi><mn>1</mn></mrow> <mi>X</mi></msubsup><msub><mo stretchy="false">|</mo> <mi>U</mi></msub><mo>∈</mo><msup><mi>Ω</mi> <mn>3</mn></msup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mu^X_{F1}|_{U} \in \Omega^3(U)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>μ</mi> <mrow><mi>F</mi><mn>1</mn></mrow> <mi>X</mi></msubsup></mrow><annotation encoding="application/x-tex">\mu_{F1}^X</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> is exact, and hence admits a potential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mi>U</mi></msub><mo>∈</mo><msup><mi>Ω</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B_U \in \Omega^2(U)</annotation></semantics></math>, i.e. such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>B</mi><mo>=</mo><msubsup><mi>μ</mi> <mrow><mi>F</mi><mn>1</mn></mrow> <mi>X</mi></msubsup><msub><mo stretchy="false">|</mo> <mi>U</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{d} B = \mu^X_{F1}|_U</annotation></semantics></math>.</p> <p>Then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> a 2-dimensional <a class="existingWikiWord" href="/nlab/show/closed+manifold">closed manifold</a>, the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+action+functional">Green-Schwarz action functional</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mi>ℏ</mi></mfrac></mstyle><msub><mi>S</mi> <mi>GS</mi></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><msub><mo stretchy="false">]</mo> <mi>U</mi></msub><mo>⟶</mo><mi>ℝ</mi><msub><mo stretchy="false">/</mo> <mi>ℏ</mi></msub><mi>ℤ</mi></mrow><annotation encoding="application/x-tex"> \exp(\tfrac{i}{\hbar} S_{GS}) \;\colon\; [\Sigma,X]_U \longrightarrow \mathbb{R}/_{\hbar} \mathbb{Z} </annotation></semantics></math></div> <p>is the function on the super-<a class="existingWikiWord" href="/nlab/show/smooth+mapping+space">smooth mapping space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><msub><mo stretchy="false">]</mo> <mi>U</mi></msub></mrow><annotation encoding="application/x-tex">[\Sigma,X]_U</annotation></semantics></math> of morphisms of <a class="existingWikiWord" href="/nlab/show/supermanifolds">supermanifolds</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><mi>Σ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\phi \colon \Sigma \to X</annotation></semantics></math> which factor through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>, given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mspace width="thickmathspace"></mspace><mo>↦</mo><mspace width="thickmathspace"></mspace><msub><mo>∫</mo> <mi>Σ</mi></msub><msub><mi>vol</mi> <mrow><msup><mi>ϕ</mi> <mo>*</mo></msup><mi>e</mi></mrow></msub><mspace width="thickmathspace"></mspace><mo>+</mo><mspace width="thickmathspace"></mspace><msub><mo>∫</mo> <mi>Σ</mi></msub><msup><mi>ϕ</mi> <mo>*</mo></msup><msub><mi>B</mi> <mi>U</mi></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>d</mi></mstyle><msub><mi>B</mi> <mi>U</mi></msub><mo>=</mo><msubsup><mi>μ</mi> <mn>3</mn> <mi>X</mi></msubsup><msub><mo stretchy="false">|</mo> <mi>U</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \phi \;\mapsto\; \int_\Sigma vol_{\phi^\ast e} \;+\; \int_\Sigma \phi^\ast B_U \;\;\;\,,\;\;\;\;\;\;\;\;\; \mathbf{d} B_U = \mu^X_3|_U \,. </annotation></semantics></math></div> <p>In order to get rid of the restriction to some <a class="existingWikiWord" href="/nlab/show/chart">chart</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">U \subset X</annotation></semantics></math> one needs to add global data. The need for this is at least mentioned briefly in (<a href="#Witten86">Witten 86, p. 261 (17 of 20)</a>), but seems to have otherwise been ignored in the physics literature. The general solution is to promote the local potentials <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> to the connection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>B</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat B</annotation></semantics></math> on a <a class="existingWikiWord" href="/nlab/show/Super+Gerbes">super gerbe</a> (<a href="#FiorenzaSatiSchreiber13">FSS 13</a>). This is a choice of <a class="existingWikiWord" href="/nlab/show/higher+prequantization">higher prequantization</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>ℝ</mi><msub><mo stretchy="false">/</mo> <mi>ℏ</mi></msub><mi>ℤ</mi><mo stretchy="false">)</mo></mtd> <mtd><mtext>prequantization</mtext></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mover><mi>B</mi><mo stretchy="false">^</mo></mover></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>curv</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><msubsup><mi>μ</mi> <mrow><mi>F</mi><mn>1</mn></mrow> <mi>X</mi></msubsup></mrow></munder></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mn>3</mn></msup></mtd> <mtd><mtext>3-form curvature</mtext></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && \mathbf{B}^{2}(\mathbb{R}/_\hbar \mathbb{Z}) & \text{prequantization} \\ & {}^{\mathllap{\hat B}}\nearrow & \downarrow^{\mathrlap{curv}} \\ X &\underset{\mu^X_{F1}}{\longrightarrow}& \mathbf{\Omega}^3 & \text{3-form curvature} } \,. </annotation></semantics></math></div> <p>Writing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∫</mo> <mi>Σ</mi></msub><msup><mi>ϕ</mi> <mo>*</mo></msup><mover><mi>B</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\int_\Sigma \phi^\ast \hat B</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/volume+holonomy">volume holonomy</a> of a <a class="existingWikiWord" href="/nlab/show/circle+2-bundle+with+connection">circle 2-bundle with connection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>B</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat B</annotation></semantics></math>, then the globally defined Green-Schwarz sigma model</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mi>ℏ</mi></mfrac></mstyle><msub><mi>S</mi> <mi>GS</mi></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo>⟶</mo><mi>ℝ</mi><msub><mo stretchy="false">/</mo> <mi>ℏ</mi></msub><mi>ℤ</mi></mrow><annotation encoding="application/x-tex"> \exp(\tfrac{i}{\hbar} S_{GS}) \;\colon\; [\Sigma, X] \longrightarrow \mathbb{R}/_\hbar\mathbb{Z} </annotation></semantics></math></div> <p>is given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mspace width="thickmathspace"></mspace><mo>↦</mo><mspace width="thickmathspace"></mspace><msub><mo>∫</mo> <mi>Σ</mi></msub><msub><mi>vol</mi> <mrow><msup><mi>ϕ</mi> <mo>*</mo></msup></mrow></msub><mo>+</mo><msub><mo>∫</mo> <mi>Σ</mi></msub><msup><mi>ϕ</mi> <mo>*</mo></msup><mover><mi>B</mi><mo stretchy="false">^</mo></mover><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>curv</mi><mo stretchy="false">(</mo><mover><mi>B</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>μ</mi> <mrow><mi>F</mi><mn>1</mn></mrow> <mi>X</mi></msubsup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \phi \;\mapsto\; \int_\Sigma vol_{\phi^\ast} + \int_\Sigma \phi^\ast \hat B \;\;\,, \;\;\;\;\;\;\; curv(\hat B) = \mu_{F1}^X \,. </annotation></semantics></math></div> <p>This form of the Green-Schwarz action functional for the string has evident generalization to other <a class="existingWikiWord" href="/nlab/show/p-branes">p-branes</a>. Whenever there is a <a class="existingWikiWord" href="/nlab/show/spin+group">Spin(d-1,1)</a>-<a class="existingWikiWord" href="/nlab/show/invariant">invariant</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(p+2)</annotation></semantics></math>-cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mrow><mi>p</mi><mo>+</mo><mn>2</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\mu_{p+2}</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mi>N</mi></mstyle></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d-1,1\vert \mathbf{N}}</annotation></semantics></math>, then one may ask for a higher gerbe (<a class="existingWikiWord" href="/nlab/show/higher+prequantum+line+bundle">higher prequantum line bundle</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>C</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat C</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>μ</mi> <mrow><mi>p</mi><mo>+</mo><mn>2</mn></mrow> <mi>X</mi></msubsup></mrow><annotation encoding="application/x-tex">\mu^X_{p+2}</annotation></semantics></math> and consider the analogous functional.</p> <p>The triples <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mstyle mathvariant="bold"><mi>N</mi></mstyle><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(d,\mathbf{N},p)</annotation></semantics></math> (spacetime dimension, number of supersymmetries, dimension of brane) such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mrow><mi>p</mi><mo>+</mo><mn>2</mn></mrow></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mover><mi>ψ</mi><mo>¯</mo></mover><mo>∧</mo><msup><mi>Γ</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub><mi>⋯</mi><msub><mi>a</mi> <mi>p</mi></msub></mrow></msup><mi>ψ</mi><mo>∧</mo><msub><mi>e</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msub><mo>∧</mo><mi>⋯</mi><mo>∧</mo><msub><mi>e</mi> <mrow><msub><mi>a</mi> <mi>p</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex"> \mu_{p+2} \;\coloneqq\; \overline{\psi} \wedge \Gamma^{a_1 \cdots a_p} \psi \wedge e_{a_1} \wedge \cdots \wedge e_{a_p} </annotation></semantics></math></div> <p>is a nontrivial cocycle, hence for which there is such a Green-Schwarz action functional for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-branes on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mi>N</mi></mstyle></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d-1,1\vert \mathbf{N}}</annotation></semantics></math> may be classified and form what is called the <em><a class="existingWikiWord" href="/nlab/show/brane+scan">brane scan</a></em> (<a href="#AETW87">Achúcarro-Evans-TownsendWiltshire 87</a>, <a href="#Brandt12-13">Brandt 12-13</a>):</p> <p id="DuffBraneScan"></p> <div style="float:left;margin:0 10px 10px 0;"> <img src="https://ncatlab.org/nlab/files/DuffBraneScan.jpg" height="600" /> </div> <p>The graphics on the left is from (<a href="#Duff87">Duff 87</a>). The diagonal lines indicate <a class="existingWikiWord" href="/nlab/show/double+dimensional+reduction">double dimensional reduction</a>, taking a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(p+1)</annotation></semantics></math>-brane in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(d+1)</annotation></semantics></math> dimensions to a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-brane in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>-dimensions.</p> <p>For instance for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>=</mo><mn>11</mn><mo>,</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>N</mi></mstyle><mo>=</mo><mstyle mathvariant="bold"><mn>32</mn></mstyle><mo>,</mo><mspace width="thickmathspace"></mspace><mi>p</mi><mo>=</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(d = 11, \; \mathbf{N} = \mathbf{32}, \; p = 2)</annotation></semantics></math> one finds a cocycle, and the corresponding GS-action functional is that of the fundamental <a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a>.</p> <p>This was a striking confluence of brane physics and classification of <a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super</a> <a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a>. But just as striking as the matching, was what it lacked to match: the <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a> and the <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><mn>11</mn></mrow><annotation encoding="application/x-tex">d = 11</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>=</mo><mn>5</mn></mrow><annotation encoding="application/x-tex">p = 5</annotation></semantics></math>) are lacking from the old brane scan. Incidentally, these lacking branes are precisely those branes on which the branes that do appear on the brane scan may end, equivalently those branes that have <a class="existingWikiWord" href="/nlab/show/higher+gauge+fields">higher gauge fields</a> on their <a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a> (tensor multiplets).</p> <p>An action functional for the <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> analogous to a Green-Schwarz action functional was found in (<a href="#BLNPST97">BLNPST 97</a>, <a href="#APPS97">APPS 97</a>). It is again the sum of a kinetic term and a WZW-like term, but the WZW-like term does not come from a cocycle on a (super-)group.</p> <p>In order to deal with this, it was suggested in (<a href="#CAIB99">CAIB 99</a>, <a href="#Sakaguchi00">Sakaguchi 00</a>, <a href="#AzcarragaIzquierdo01">Azcarraga-Izquierdo 01</a>) that there is an algebraic structure called “<a class="existingWikiWord" href="/nlab/show/extended+super-Minkowski+spacetimes">extended super-Minkowski spacetimes</a>” that generalizes <a class="existingWikiWord" href="/nlab/show/super+Minkowski+spacetime">super Minkowski spacetime</a> and serves to unify the Green-Schwarz-like models for the D-branes and the M5-brane with the original Green-Schwarz models for the string and the M2-brane.</p> <p>These <a class="existingWikiWord" href="/nlab/show/extended+super-Minkowski+spacetimes">extended super-Minkowski spacetimes</a> carry algebraic analogs of super Lie algebra cocycles, such that the relevant terms for the <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a> and the <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> do appear after all, hence such that all the branes in <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a>/<a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a> are unified. In fact these “<a class="existingWikiWord" href="/nlab/show/extended+super-Minkowski+spacetimes">extended super-Minkowski spacetimes</a>” are precisely the “FDA”s that have been introduced before in the <a class="existingWikiWord" href="/nlab/show/D%27Auria-Fr%C3%A9+formulation+of+supergravity">D'Auria-Fré formulation of supergravity</a> and what became identified as the 7-cocycle for the <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> this way had earlier been recognized algebraically as an stepping stone for an elegant re-derivation of <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a> (<a href="#DAuriaFre82">D’Auria-Fré 82</a>).</p> <p>The (<a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher</a>) geometric meaning of these constructions was found in (<a href="#FiorezaSatiSchreiber13">Fiorenza-Sati-Schreiber 13</a>): these algebraic structures of “<a class="existingWikiWord" href="/nlab/show/extended+super-Minkowski+spacetimes">extended super-Minkowski spacetimes</a>”/FDAs are precisely the <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebras">Chevalley-Eilenberg algebras</a> of <a class="existingWikiWord" href="/nlab/show/super+Lie+n-algebra">super Lie n-algebra</a>-extensions of <a class="existingWikiWord" href="/nlab/show/super-Minkowski+spacetime">super-Minkowski spacetime</a> which are classified by the cocycles that serve as the GS-WZW terms of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">p_1</annotation></semantics></math>-branes that may end on those <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">p_2</annotation></semantics></math>-branes whose cocycles are carried by the extended super-Minkowski spacetime.</p> <p>Hence the missing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-branes in the old <a class="existingWikiWord" href="/nlab/show/brane+scan">brane scan</a> (classifying just cocycles on <a class="existingWikiWord" href="/nlab/show/super+Lie+algebras">super Lie algebras</a>) do appear as one generalizes (super) Lie algebras to (super) <a class="existingWikiWord" href="/nlab/show/strong+homotopy+Lie+algebras">strong homotopy Lie algebras</a> = <a class="existingWikiWord" href="/nlab/show/L-infinity+algebras">L-infinity algebras</a>. Moreover, each brane intersection law (one brane species may end on another) is now matched to a super <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra <a class="existingWikiWord" href="/nlab/show/extension">extension</a> and so the old brane scan is generalized to a tree of branes <em><a class="existingWikiWord" href="/schreiber/show/The+brane+bouquet">The brane bouquet</a></em>:</p> <p><img src="https://ncatlab.org/schreiber/files/BraneBouquetWithDualities.jpg" alt="the brane bouquet" width="900" /></p> <p>Each item in this bouquet denotes a <a class="existingWikiWord" href="/nlab/show/super+L-infinity+algebra">super L-infinity algebra</a> and each arrow denotes an <a class="existingWikiWord" href="/nlab/show/L-infinity+extension">L-infinity extension</a> classified by a cocycle which encodes the GS-WZW term of the brane named by the domain of the arrow. Moreover, arrows pass exactly from one brane species to the brane species that may end on the former.</p> <p>In (<a href="#FiorenzaSatiSchreiber13">Fiorenza-Sati-Schreiber 13</a>) it is shown that all these <a class="existingWikiWord" href="/nlab/show/super+L-infinity+algebras">super L-infinity algebras</a> <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integrate</a> to <a class="existingWikiWord" href="/nlab/show/smooth+super-n-groups">smooth super-n-groups</a>, and all the cocycles <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integrate</a> to <a class="existingWikiWord" href="/nlab/show/Super+Gerbes">super-gerbes</a> on these, such that the induced <a class="existingWikiWord" href="/nlab/show/volume+holonomy">volume holonomy</a> is the relevant generalized GS-WZW term. For detailed exposition see at <em><a class="existingWikiWord" href="/schreiber/show/Structure+Theory+for+Higher+WZW+Terms">Structure Theory for Higher WZW Terms</a></em>.</p> <p>With this generalized perspective, now the Green-Schwarz-type action functionals describe <em>all</em> the <a class="existingWikiWord" href="/nlab/show/p-branes">p-branes</a> in <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a>/<a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a>.</p> <p>Again, in order to make this generally true one needs to apply a <a class="existingWikiWord" href="/nlab/show/higher+prequantization">higher prequantization</a> – a choice of <a class="existingWikiWord" href="/nlab/show/line+n-bundle+with+connection">line (p+1)-bundle with connection</a> – in order to globalize the <a class="existingWikiWord" href="/nlab/show/WZW-terms">WZW-terms</a> (<a href="#FiorenzaSatiSchreiber13">Fiorenza-Sati-Schreiber 13</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>ℝ</mi><msub><mo stretchy="false">/</mo> <mi>ℏ</mi></msub><mi>ℤ</mi><mo stretchy="false">)</mo></mtd> <mtd><mtext>prequantization</mtext></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mover><mi>A</mi><mo stretchy="false">^</mo></mover> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>curv</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><msubsup><mi>μ</mi> <mrow><mi>p</mi><mo>+</mo><mn>2</mn></mrow> <mi>X</mi></msubsup></mrow></munder></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mrow><mi>p</mi><mo>+</mo><mn>2</mn></mrow></msup></mtd> <mtd><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo><mtext>-form curvature</mtext></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && \mathbf{B}^{p+1}(\mathbb{R}/_\hbar \mathbb{Z}) & \text{prequantization} \\ & {}^{\mathllap{\hat A_{p+1}}}\nearrow & \downarrow^{\mathrlap{curv}} \\ X &\underset{\mu^X_{p+2}}{\longrightarrow}& \mathbf{\Omega}^{p+2} & (p+2)\text{-form curvature} } \,. </annotation></semantics></math></div> <p>Hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>A</mi><mo stretchy="false">^</mo></mover> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\hat A_{p+1}</annotation></semantics></math> is the actual <a class="existingWikiWord" href="/nlab/show/background+field">background field</a> that the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-brane couples to. There is considerably more information in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>A</mi><mo stretchy="false">^</mo></mover> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">\hat A_p</annotation></semantics></math> than in its curvature <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>curv</mi><mo stretchy="false">(</mo><msub><mover><mi>A</mi><mo stretchy="false">^</mo></mover> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>μ</mi> <mrow><mi>p</mi><mo>+</mo><mn>2</mn></mrow></msub></mrow><annotation encoding="application/x-tex">curv(\hat A_{p+1}) = \mu_{p+2}</annotation></semantics></math>. For instance for the <a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a> one may find the local super <a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a> for local choices of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>A</mi><mo stretchy="false">^</mo></mover> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\hat A_{p+1}</annotation></semantics></math> for the given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">\mu_{4}</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/KK-compactifications">KK-compactifications</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">d = 4</annotation></semantics></math>. It turns out that the bosonic <a class="existingWikiWord" href="/nlab/show/body">body</a> of this moduli space is the <a class="existingWikiWord" href="/nlab/show/exceptional+tangent+bundle">exceptional tangent bundle</a> on which the <a class="existingWikiWord" href="/nlab/show/U-duality">U-duality</a> group <a class="existingWikiWord" href="/nlab/show/E7">E7</a> has a canonical action (see at <em><a class="existingWikiWord" href="/schreiber/show/From+higher+to+exceptional+geometry">From higher to exceptional geometry</a></em>).</p> <p>This highlights that Green-Schwarz functionals capture fundamental (“microscopic”) aspects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-branes. In contrast, often <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-branes are discussed in their <a class="existingWikiWord" href="/nlab/show/soliton">solitonic</a> incarnation as <em><a class="existingWikiWord" href="/nlab/show/black+branes">black branes</a></em>. These solitonic branes sit at asymptotic boundaries of <a class="existingWikiWord" href="/nlab/show/anti-de+Sitter+spacetime">anti-de Sitter spacetime</a> and carry <a class="existingWikiWord" href="/nlab/show/conformal+field+theories">conformal field theories</a>, related to the ambient <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> by <a class="existingWikiWord" href="/nlab/show/AdS-CFT+duality">AdS-CFT duality</a>.</p> <p>This phenomenon is indeed a consequence of the fundamental Green-Schwarz branes:</p> <p>Consider a 1/2-<a class="existingWikiWord" href="/nlab/show/BPS+state">BPS state</a> solution of <a class="existingWikiWord" href="/nlab/show/type+II+supergravity">type II supergravity</a> or <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a>, respectively. These solutions locally happen to have the same classification as the Green-Schwarz branes. Hence we may consider a configuration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><mi>Σ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\phi \colon \Sigma \to X</annotation></semantics></math> of the corresponding fundamental <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-brane which embeds <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> into the asymptotic AdS boundary of the given 1/2 BPS spacetime <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. Then it turns out that restricting the Green-Schwarz action functional to small fluctuations around this configuration, and applying a <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphism</a> <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixing</a>, then the resulting action functional is that of a <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetric</a> <a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> as in the <a class="existingWikiWord" href="/nlab/show/AdS-CFT">AdS-CFT</a> dictionary:</p> <table><thead><tr><th>fundamental <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-brane</th><th>-fluctuations about asymptotic AdS configuration<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></th><th>solitonic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-brane</th></tr></thead><tbody><tr><td style="text-align: left;">Green-Schwarz action functional</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/supersymmetric">super</a>-<a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a></td></tr> </tbody></table> <p>(<a href="#ClausKalloshProeyen97">Claus-Kallosh-Proeyen 97</a>, <a href="#ClausKalloshKumarTownsend98">Claus-Kallosh-Kumar-Townsend 98</a>, <a href="#AFFFTT98">AFFFTT 98</a> <a href="#PastiSorokinTonin99">Pasti-Sorokin-Tonin 99</a>)</p> <p>In fact the <a class="existingWikiWord" href="/nlab/show/BPS-state">BPS-state</a> condition itself is neatly encoded in the Green-Schwarz action functionals: by construction they are invariant under the <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <a class="existingWikiWord" href="/nlab/show/superisometry+group">superisometry group</a>. Hence the <a class="existingWikiWord" href="/nlab/show/Noether+theorem">Noether theorem</a> implies that there are corresponding <a class="existingWikiWord" href="/nlab/show/conserved+currents">conserved currents</a>, whose <a class="existingWikiWord" href="/nlab/show/Dickey+bracket">Dickey bracket</a> forms a super-<a class="existingWikiWord" href="/nlab/show/Lie+algebra+extension">Lie algebra extension</a> of the Lie algebra of supersymmetries.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mo>=</mo></mover></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mover><mi>C</mi><mo stretchy="false">^</mo></mover></mpadded></msub><mo>↘</mo></mtd> <mtd><mo>⇙</mo></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mover><mi>C</mi><mo stretchy="false">^</mo></mover></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>ℝ</mi><msub><mo stretchy="false">/</mo> <mi>ℏ</mi></msub><mi>ℤ</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>}</mo></mrow></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mo>≃</mo></mover></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mover><mi>C</mi><mo stretchy="false">^</mo></mover></mpadded></msub><mo>↘</mo></mtd> <mtd><mo>⇙</mo></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mover><mi>C</mi><mo stretchy="false">^</mo></mover></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>ℝ</mi><msub><mo stretchy="false">/</mo> <mi>ℏ</mi></msub><mi>ℤ</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>}</mo></mrow></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mo>≃</mo></mover></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow><mo>}</mo></mrow></mtd></mtr> <mtr><mtd><mtext>topological currents</mtext></mtd> <mtd></mtd> <mtd><mtext>Noether currents</mtext></mtd> <mtd></mtd> <mtd><mtext>symmetries</mtext></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \left\{ \array{ X && \overset{=}{\longrightarrow} && X \\ & {}_{\mathllap{\hat C}}\searrow &\swArrow& \swarrow_{\mathrlap{\hat C}} \\ && \mathbf{B}^{p+1}(\mathbb{R}/_{\hbar} \mathbb{Z}) } \right\} &\longrightarrow& \left\{ \array{ X && \overset{\simeq}{\longrightarrow} && X \\ & {}_{\mathllap{\hat C}}\searrow &\swArrow& \swarrow_{\mathrlap{\hat C}} \\ && \mathbf{B}^{p+1}(\mathbb{R}/_{\hbar} \mathbb{Z}) } \right\} &\longrightarrow& \left\{ \array{ X && \overset{\simeq}{\longrightarrow} && X } \right\} \\ \text{topological currents} && \text{Noether currents} && \text{symmetries} } </annotation></semantics></math></div> <p>Here the “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⇙</mo></mrow><annotation encoding="application/x-tex">\swArrow</annotation></semantics></math>” filling the triangles is the non-trivial gauge transformation by which the <a class="existingWikiWord" href="/nlab/show/WZW+term">WZW term</a> (as any WZW term) is preserved under the symmetries (instead of being fixed identically). It is the information in this transformations which makes the currents form an extension of the symmetries.</p> <p>Here this yields the famous brane charge extensions of the super-isometry super Lie algebra of the schematic form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>Q</mi> <mi>α</mi></msub><mo>,</mo><msub><mi>Q</mi> <mi>β</mi></msub><mo stretchy="false">}</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mi>C</mi><msubsup><mi>Γ</mi> <mrow><mi>α</mi><mi>β</mi></mrow> <mi>a</mi></msubsup><mo stretchy="false">)</mo><msub><mi>P</mi> <mi>a</mi></msub><mspace width="thickmathspace"></mspace><mo>+</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mi>C</mi><msup><mi>Γ</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub><mi>⋯</mi><msub><mi>a</mi> <mi>p</mi></msub></mrow></msup><msub><mo stretchy="false">)</mo> <mrow><mi>α</mi><mi>β</mi></mrow></msub><msub><mi>Z</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>a</mi> <mi>p</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex"> \{Q_\alpha, Q_\beta\} \;=\; (C \Gamma^a_{\alpha \beta}) P_a \;+\; (C \Gamma^{a_1 \cdots a_p})_{\alpha \beta} Z_{a_1, \cdots, a_p} </annotation></semantics></math></div> <p>(for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/Killing+spinor">Killing spinor</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> its corresponding <a class="existingWikiWord" href="/nlab/show/Killing+vector">Killing vector</a>) known as the <a class="existingWikiWord" href="/nlab/show/type+II+supersymmetry+algebra">type II supersymmetry algebra</a> and the <a class="existingWikiWord" href="/nlab/show/M-theory+supersymmetry+algebra">M-theory supersymmetry algebra</a>, respectively (<a href="#AGIT89">Azcárraga-Gauntlett-Izquierdo-Townsend 89</a>). In fact it yields super-<a class="existingWikiWord" href="/nlab/show/L-infinity+algebra+extensions">Lie n-algebra extensions</a> of which the familiar super Lie algebra extensions are the 0-truncation (<a href="#SatiSchreiber15">Sati-Schreiber 15</a>, <a class="existingWikiWord" href="/schreiber/show/Local+prequantum+field+theory">Khavkine-Schreiber 16</a>).</p> <p>In summary, the nature and classification of Green-Schwarz action functionals captures in a mathematically precise way a good deal of the core structure of <a class="existingWikiWord" href="/nlab/show/string+theory">string</a>/<a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a>.</p> <p>In fact, the <a class="existingWikiWord" href="/nlab/show/super+L-infinity+algebra">super Lie-n algebraic</a> perspective on the Green-Schwarz functionals via the <a class="existingWikiWord" href="/schreiber/show/brane+bouquet">brane bouquet</a> also solves the following open problem on <a class="existingWikiWord" href="/nlab/show/M-branes">M-branes</a>:</p> <p>it is famously known from <a class="existingWikiWord" href="/nlab/show/Freed-Witten+anomaly">Freed-Witten anomaly</a>-cancellation that the <a class="existingWikiWord" href="/nlab/show/D-brane+charges">D-brane charges</a> are not in fact just in <a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a> in every second degree, but are in <a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a>, hence <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rationally</a> in <a class="existingWikiWord" href="/nlab/show/twisted+de+Rham+cohomology">twisted de Rham cohomology</a>, with the twist being the <a class="existingWikiWord" href="/nlab/show/F1-brane">F1-brane</a> charge (from the fundamental). It is an open problem to determine what becomes of these <a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a> charge groups as one lifts F1/D<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-branes in string theory to M2/M5-branes in <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a>.</p> <table><thead><tr><th></th><th>intersecting branes</th><th>charges in <a class="existingWikiWord" href="/nlab/show/generalized+cohomology+theory">generalized cohomology theory</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/F1-brane">F1</a>/<a class="existingWikiWord" href="/nlab/show/D-brane">Dp-branes</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M2-brane">M2</a>/<a class="existingWikiWord" href="/nlab/show/M5-branes">M5-branes</a></td><td style="text-align: left;">???</td></tr> </tbody></table> <p>Notice that there are “microscopic degrees of freedom” of the theory encoded by the choice of <a class="existingWikiWord" href="/nlab/show/generalized+cohomology+theory">generalized cohomology theory</a> here, generalizing the extra degrees of freedom in the choice of a WZW-term already mentioned above. In general for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> a cohomology theory and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>⟶</mo><mi>E</mi><mo>⊗</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">E \longrightarrow E \otimes \mathbb{Q}</annotation></semantics></math> its <a class="existingWikiWord" href="/nlab/show/Chern+character">Chern character</a> map (for instance from <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a> to <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> in every second degree), then a choice of genuine charges is the extra information encoded in a lift</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>E</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mtext>true charge</mtext></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>ch</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><munder><mo>⟶</mo><mfrac linethickness="0"><mrow><mtext>rational</mtext></mrow><mrow><mtext>charge</mtext></mrow></mfrac></munder></mtd> <mtd><mi>E</mi><mo>⊗</mo><mi>ℚ</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && E \\ & {}^{\mathllap{\text{true charge}}}\nearrow & \downarrow^{\mathrlap{ch}} \\ X &\underset{\text{rational} \atop \text{charge}}{\longrightarrow}& E \otimes \mathbb{Q} } </annotation></semantics></math></div> <p>But <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rationally</a> <a class="existingWikiWord" href="/schreiber/show/The+brane+bouquet">The brane bouquet</a> allows to derive this from first principles:</p> <p>Above we saw that the naive cocycles of the <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a> and of the <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> are not defined on the actual <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>, but on some “extended” spacetime, which is really a <a class="existingWikiWord" href="/nlab/show/smooth+super+infinity-groupoid">smooth super infinity-groupoid</a> extension of spacetime. Hence we should ask if these cocycles <a class="existingWikiWord" href="/nlab/show/descent">descend</a> to the actual super-spacetime while picking up some twists.</p> <p>One may prove that:</p> <ul> <li> <p>the F1/D<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-brane GS-WZW cocycles descend to 10d type II superspacetime to form a single cocycle in rational twisted K-theory, just as the traditional lore reqires (<a href="#FiorenzaSatiSchreiber16">Fiorenza-Sati-Schreiber 16</a>);</p> </li> <li> <p>the M2/M5 GS-WZW cocycles descent to 11d superspacetime to form a single cocycle with values in the <a class="existingWikiWord" href="/nlab/show/rational+n-sphere">rational 4-sphere</a> (<a href="#FiorenzaSatiSchreiber16">Fiorenza-Sati-Schreiber 16</a>).</p> </li> </ul> <p>This has implications on some open conjectures regarding <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a>, for more on this see <em><a class="existingWikiWord" href="/schreiber/show/Equivariant+cohomology+of+M2%2FM5-branes">Equivariant cohomology of M2/M5-branes</a></em>.</p> <h2 id="definition">Definition</h2> <p>The Green-Schwarz action functionals are of the standard <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> form for <a class="existingWikiWord" href="/nlab/show/target+spaces">target spaces</a> that are <a class="existingWikiWord" href="/nlab/show/supermanifold">super</a>-<a class="existingWikiWord" href="/nlab/show/homogeneous+spaces">homogeneous spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">G/H</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/Lie+supergroup">Lie supergroup</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> a sub-super-group, and for <a class="existingWikiWord" href="/nlab/show/background+gauge+fields">background gauge fields</a> that are super-<a class="existingWikiWord" href="/nlab/show/WZW+model">WZW</a>-<a class="existingWikiWord" href="/nlab/show/circle+n-bundles+with+connection">circle n-bundles with connection</a>/<a class="existingWikiWord" href="/nlab/show/bundle+gerbes">bundle gerbes</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>.</p> <ul> <li>for branes on <a class="existingWikiWord" href="/nlab/show/super+Minkowski+spacetime">super Minkowski spacetime</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/super+Poincar%C3%A9+Lie+algebra">super Poincaré Lie algebra</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔥</mi></mrow><annotation encoding="application/x-tex">\mathfrak{h}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/Lorentz+Lie+algebra">Lorentz Lie algebra</a>;</li> </ul> <p>These action functionals were first considered in (<a href="#GreenSchwarz84">Green-Schwarz 84</a>) for <a class="existingWikiWord" href="/nlab/show/superstrings">superstrings</a> in various dimensions. The full interpretation of the action functional as an higher <a class="existingWikiWord" href="/nlab/show/Wess-Zumino-Witten+theory">Wess-Zumino-Witten theory</a>-type action controled by the <a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a> of the <a class="existingWikiWord" href="/nlab/show/super+Poincar%C3%A9+Lie+algebra">super Poincaré Lie algebra</a> (or rather of the <a class="existingWikiWord" href="/nlab/show/super+translation+Lie+algebra">super translation Lie algebra</a> inside it) is due to (<a href="#AzcarragaTownsend89">Azcárraga-Townsend89</a>).</p> <ul> <li>for branes on <a class="existingWikiWord" href="/nlab/show/super+anti+de+Sitter+spacetime">super anti de Sitter spacetime</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/superconformal+group">superconformal group</a> (e.g. <a href="#MetsaevTseytlin98">Metsaev-Tseytlin 98, section 3</a>)</li> </ul> <h3 id="Supercoordinates">Supercoordinates</h3> <p>We briefly review some basics of the canonical <a class="existingWikiWord" href="/nlab/show/coordinates">coordinates</a> and the super <a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a> of the <a class="existingWikiWord" href="/nlab/show/super+Poincar%C3%A9+Lie+algebra">super Poincaré Lie algebra</a> and <a class="existingWikiWord" href="/nlab/show/super+Minkowski+space">super Minkowski space</a>, which are referred to below (see for instance <a href="#AzcarragaTownsend89">Azcárraga-Townsend 89</a>, and see at <em><a class="existingWikiWord" href="/nlab/show/super+Cartesian+space">super Cartesian space</a></em> and at <em><a class="existingWikiWord" href="/nlab/show/signs+in+supergeometry">signs in supergeometry</a></em>.).</p> <p>By the general discussion at <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a>, we may characterize the <a class="existingWikiWord" href="/nlab/show/super+Poincar%C3%A9+Lie+algebra">super Poincaré Lie algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔰𝔦𝔰𝔬</mi><mo stretchy="false">(</mo><mi>D</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{siso}(D-1,1)</annotation></semantics></math> by its CE-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔰𝔦𝔰𝔬</mi><mo stretchy="false">(</mo><mi>D</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CE(\mathfrak{siso}(D-1,1))</annotation></semantics></math> “of <a class="existingWikiWord" href="/nlab/show/left-invariant+1-forms">left-invariant 1-forms</a>” on its group manifold.</p> <div class="num_defn" id="CEAlgebraOfSuperPoincare"> <h6 id="definition_2">Definition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔰𝔦𝔰𝔬</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CE(\mathfrak{siso}(d-1,1))</annotation></semantics></math> is generated on</p> <ul> <li> <p>elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>e</mi> <mi>a</mi></msup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{e^a\}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>ω</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\omega^{ a b}\}</annotation></semantics></math> of degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mi>even</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1,even)</annotation></semantics></math></p> </li> <li> <p>and elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>ψ</mi> <mi>α</mi></msup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\psi^\alpha\}</annotation></semantics></math> of degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mi>odd</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1,odd)</annotation></semantics></math></p> </li> </ul> <p>with the differential defined by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>CE</mi></msub><msup><mi>ω</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msup><mo>=</mo><msup><mi>ω</mi> <mi>a</mi></msup><msub><mrow></mrow> <mi>b</mi></msub><mo>∧</mo><msup><mi>ω</mi> <mrow><mi>b</mi><mi>c</mi></mrow></msup></mrow><annotation encoding="application/x-tex"> d_{CE} \omega^{a b} = \omega^a{}_b \wedge \omega^{b c} </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>CE</mi></msub><msup><mi>e</mi> <mi>a</mi></msup><mo>=</mo><msup><mi>ω</mi> <mi>a</mi></msup><msub><mrow></mrow> <mi>b</mi></msub><mo>∧</mo><msup><mi>e</mi> <mi>b</mi></msup><mo>+</mo><mfrac><mi>i</mi><mn>2</mn></mfrac><mover><mi>ψ</mi><mo stretchy="false">¯</mo></mover><msup><mi>Γ</mi> <mi>a</mi></msup><mi>ψ</mi></mrow><annotation encoding="application/x-tex"> d_{CE} e^{a } = \omega^a{}_b \wedge e^b + \frac{i}{2}\bar \psi \Gamma^a \psi </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>CE</mi></msub><mi>ψ</mi><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msup><mi>ω</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msup><msub><mi>Γ</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msub><mi>ψ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> d_{CE} \psi = \frac{1}{4} \omega^{ a b} \Gamma_{a b} \psi \,. </annotation></semantics></math></div> <p>Removing the terms involving <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> here this is the <a class="existingWikiWord" href="/nlab/show/super+translation+algebra">super translation algebra</a>.</p> </div> <p>In this way the <a class="existingWikiWord" href="/nlab/show/super-Poincar%C3%A9+Lie+algebra">super-Poincaré Lie algebra</a> and its extensions is usefully discussed for instance in (<a href="#DAuriaFre82">D’Auria-Fré 82</a>) and in (<a href="#AzcarragaTownsend89">Azcárraga-Townsend 89</a>, <a href="#CAIB99">CAIB 99</a>). In much of the literature instead the following equivalent notation is popular, which more explicitly involves the <a class="existingWikiWord" href="/nlab/show/coordinates">coordinates</a> on <a class="existingWikiWord" href="/nlab/show/super+Minkowski+space">super Minkowski space</a>.</p> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>The abstract generators in def. <a class="maruku-ref" href="#CEAlgebraOfSuperPoincare"></a> are identified with <a class="existingWikiWord" href="/nlab/show/left+invariant+1-forms">left invariant 1-forms</a> on the <a class="existingWikiWord" href="/nlab/show/super-translation+group">super-translation group</a> (= <a class="existingWikiWord" href="/nlab/show/super+Minkowski+space">super Minkowski space</a>) as follows.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>x</mi> <mi>a</mi></msup><mo>,</mo><msup><mi>θ</mi> <mi>α</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x^a, \theta^\alpha)</annotation></semantics></math> be the canonical <a class="existingWikiWord" href="/nlab/show/coordinates">coordinates</a> on the <a class="existingWikiWord" href="/nlab/show/supermanifold">supermanifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo stretchy="false">|</mo><mi>N</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d|N}</annotation></semantics></math> underlying the super translation group. Then the identification is</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ψ</mi> <mi>α</mi></msup><mo>=</mo><mi>d</mi><msup><mi>θ</mi> <mi>α</mi></msup></mrow><annotation encoding="application/x-tex">\psi^\alpha = d \theta^\alpha</annotation></semantics></math>.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>e</mi> <mi>a</mi></msup><mo>=</mo><mi>d</mi><msup><mi>x</mi> <mi>a</mi></msup><mo>+</mo><mfrac><mi>i</mi><mn>2</mn></mfrac><mover><mi>θ</mi><mo>¯</mo></mover><msup><mi>Γ</mi> <mi>a</mi></msup><mi>d</mi><mi>θ</mi></mrow><annotation encoding="application/x-tex">e^a = d x^a + \frac{i}{2} \overline{\theta} \Gamma^a d \theta</annotation></semantics></math>.</p> </li> </ul> <p>Notice that this then gives the above formula for the differential of the <a class="existingWikiWord" href="/nlab/show/super-vielbein">super-vielbein</a> in def. <a class="maruku-ref" href="#CEAlgebraOfSuperPoincare"></a> as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>d</mi><msup><mi>e</mi> <mi>a</mi></msup></mtd> <mtd><mo>=</mo><mi>d</mi><mo stretchy="false">(</mo><mi>d</mi><msup><mi>x</mi> <mi>a</mi></msup><mo>+</mo><mfrac><mi>i</mi><mn>2</mn></mfrac><mover><mi>θ</mi><mo>¯</mo></mover><msup><mi>Γ</mi> <mi>a</mi></msup><mi>d</mi><mi>θ</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mfrac><mi>i</mi><mn>2</mn></mfrac><mi>d</mi><mover><mi>θ</mi><mo>¯</mo></mover><msup><mi>Γ</mi> <mi>a</mi></msup><mi>d</mi><mi>θ</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mfrac><mi>i</mi><mn>2</mn></mfrac><mover><mi>ψ</mi><mo>¯</mo></mover><msup><mi>Γ</mi> <mi>a</mi></msup><mi>ψ</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} d e^a & = d (d x^a + \frac{i}{2} \overline{\theta} \Gamma^a d \theta) \\ & = \frac{i}{2} d \overline{\theta}\Gamma^a d \theta \\ & = \frac{i}{2} \overline{\psi}\Gamma^a \psi \end{aligned} \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>The term <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mi>i</mi><mn>2</mn></mfrac><mover><mi>ψ</mi><mo stretchy="false">¯</mo></mover><msup><mi>Γ</mi> <mi>a</mi></msup><mi>ψ</mi></mrow><annotation encoding="application/x-tex">\frac{i}{2}\bar \psi \Gamma^a \psi</annotation></semantics></math> is sometimes called the <em><a class="existingWikiWord" href="/nlab/show/supertorsion">supertorsion</a></em> of the <a class="existingWikiWord" href="/nlab/show/super-vielbein">super-vielbein</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math>, because the defining equation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>CE</mi></msub><msup><mi>e</mi> <mi>a</mi></msup><mo>−</mo><msup><mi>ω</mi> <mi>a</mi></msup><msub><mrow></mrow> <mi>b</mi></msub><mo>∧</mo><msup><mi>e</mi> <mi>b</mi></msup><mo>=</mo><mfrac><mi>i</mi><mn>2</mn></mfrac><mover><mi>ψ</mi><mo stretchy="false">¯</mo></mover><msup><mi>Γ</mi> <mi>a</mi></msup><mi>ψ</mi></mrow><annotation encoding="application/x-tex"> d_{CE} e^{a } -\omega^a{}_b \wedge e^b = \frac{i}{2}\bar \psi \Gamma^a \psi </annotation></semantics></math></div> <p>may be read as saying that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/torsion">torsion</a>-free except for that term. Notice that this term is the only one that appears when the differential is applied to “Lorentz scalars”, hence to object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔰𝔦𝔰𝔬</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CE(\mathfrak{siso})</annotation></semantics></math> which have “all indices contracted”. (See also at <em><a class="existingWikiWord" href="/nlab/show/torsion+constraints+in+supergravity">torsion constraints in supergravity</a></em>.)</p> <p>Notably we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><mrow><mo>(</mo><mover><mi>ψ</mi><mo>¯</mo></mover><mo>∧</mo><msup><mi>Γ</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub><mi>⋯</mi><msub><mi>a</mi> <mi>p</mi></msub></mrow></msup><mi>ψ</mi><mo>∧</mo><msub><mi>e</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msub><mo>∧</mo><mi>⋯</mi><mo>∧</mo><msub><mi>e</mi> <mrow><msub><mi>a</mi> <mi>p</mi></msub></mrow></msub><mo>)</mo></mrow><mo>∝</mo><mrow><mo>(</mo><mover><mi>ψ</mi><mo>¯</mo></mover><mo>∧</mo><msup><mi>Γ</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub><mi>⋯</mi><msub><mi>a</mi> <mi>p</mi></msub></mrow></msup><mi>ψ</mi><mo>∧</mo><msub><mi>e</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msub><mo>∧</mo><mi>⋯</mi><mo>∧</mo><msub><mi>e</mi> <mrow><msub><mi>a</mi> <mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></msub><mo>)</mo></mrow><mo>∧</mo><mrow><mo>(</mo><mover><mi>Ψ</mi><mo>¯</mo></mover><mo>∧</mo><msub><mi>Γ</mi> <mrow><msub><mi>a</mi> <mi>p</mi></msub></mrow></msub><mi>Ψ</mi><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> d \left( \overline{\psi} \wedge \Gamma^{a_1 \cdots a_p} \psi \wedge e_{a_1} \wedge \cdots \wedge e_{a_p} \right) \propto \left( \overline{\psi} \wedge \Gamma^{a_1 \cdots a_p} \psi \wedge e_{a_1} \wedge \cdots \wedge e_{a_{p-1}} \right) \wedge \left( \overline{\Psi} \wedge \Gamma_{a_p} \Psi \right) \,. </annotation></semantics></math></div> <p>This remaining operation “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo>↦</mo><msup><mi>Ψ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">e \mapsto \Psi^2</annotation></semantics></math>” of the differential acting on Loretz scalars is sometimes denoted “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>t</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">t_0</annotation></semantics></math>”, e.g. in (<a href="#BossardHoweStelle09">Bossard-Howe-Stelle 09, equation (8)</a>).</p> <p>This relation is what govers all of the exceptional super Lie algebra cocycles that appear as WZW terms for the Green-Schwarz action below: for some combinations of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>D</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(D,p)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/Fierz+identity">Fierz identity</a> implies that the term</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mover><mi>ψ</mi><mo>¯</mo></mover><mo>∧</mo><msup><mi>Γ</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub><mi>⋯</mi><msub><mi>a</mi> <mi>p</mi></msub></mrow></msup><mi>ψ</mi><mo>∧</mo><msub><mi>e</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msub><mo>∧</mo><mi>⋯</mi><mo>∧</mo><msub><mi>e</mi> <mrow><msub><mi>a</mi> <mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></msub><mo>)</mo></mrow><mo>∧</mo><mrow><mo>(</mo><mover><mi>Ψ</mi><mo>¯</mo></mover><mo>∧</mo><msub><mi>Γ</mi> <mrow><msub><mi>a</mi> <mi>p</mi></msub></mrow></msub><mi>Ψ</mi><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \left( \overline{\psi} \wedge \Gamma^{a_1 \cdots a_p} \psi \wedge e_{a_1} \wedge \cdots \wedge e_{a_{p-1}} \right) \wedge \left( \overline{\Psi} \wedge \Gamma_{a_p} \Psi \right) </annotation></semantics></math></div> <p>vanishes identically, and hence in these dimensions the term</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>ψ</mi><mo>¯</mo></mover><mo>∧</mo><msup><mi>Γ</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub><mi>⋯</mi><msub><mi>a</mi> <mi>p</mi></msub></mrow></msup><mi>ψ</mi><mo>∧</mo><msub><mi>e</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msub><mo>∧</mo><mi>⋯</mi><mo>∧</mo><msub><mi>e</mi> <mrow><msub><mi>a</mi> <mi>p</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex"> \overline{\psi} \wedge \Gamma^{a_1 \cdots a_p} \psi \wedge e_{a_1} \wedge \cdots \wedge e_{a_p} </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a>. See also the <a href="#BraneScan">brane scan</a> table below.</p> </div> <h3 id="kinetic_term">Kinetic term</h3> <p>(…)</p> <p><a class="existingWikiWord" href="/nlab/show/kinetic+action">kinetic action</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∫</mo> <mi>Σ</mi></msub><mo stretchy="false">⟨</mo><msup><mi>ϕ</mi> <mo>*</mo></msup><msup><mi>Π</mi> <mi>a</mi></msup><mo>,</mo><msup><mi>ϕ</mi> <mo>*</mo></msup><msup><mi>Π</mi> <mi>b</mi></msup><msub><mi>η</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msub><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex"> \int_\Sigma \langle \phi^\ast\Pi^a, \phi^\ast \Pi^b \eta_{a b}\rangle </annotation></semantics></math></div> <p>(…)</p> <h3 id="DefinitionWZWTerm">WZW term</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>e</mi> <mi>a</mi></msup><mo>,</mo><msup><mi>ω</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msup><mo>,</mo><msup><mi>ψ</mi> <mi>α</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(e^a, \omega^{a b}, \psi^\alpha)</annotation></semantics></math> be the standard generators of the <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔰𝔦𝔰𝔬</mi><mo stretchy="false">(</mo><mi>d</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CE(\mathfrak{siso}(d,1))</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/super+Poincar%C3%A9+Lie+algebra">super Poincaré Lie algebra</a>, as discussed there.</p> <p>The part of the <a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a> of the super translation Lie algebra that is invariant under the <a class="existingWikiWord" href="/nlab/show/special+orthogonal+Lie+algebra">Lorentz</a> transformations is spanned by closed elements of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo>=</mo><mo stretchy="false">(</mo><mi>d</mi><mover><mi>θ</mi><mo stretchy="false">¯</mo></mover><msub><mi>Γ</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>a</mi> <mi>p</mi></msub></mrow></msub><mo>∧</mo><mi>d</mi><mi>θ</mi><mo stretchy="false">)</mo><mo>∧</mo><msup><mi>Π</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msup><mo>∧</mo><mi>⋯</mi><mo>∧</mo><msup><mi>Π</mi> <mrow><msub><mi>a</mi> <mi>p</mi></msub></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mu = (d \bar \theta \Gamma_{a_1, \cdots, a_p} \wedge d \theta) \wedge \Pi^{a_1} \wedge \cdots \wedge \Pi^{a_p} \,. </annotation></semantics></math></div> <p>These exist (are closed) only for certain combinations of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>. The possible values are listed <a href="#BraneScan">below</a>.</p> <p>For a bosonic <a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a> the <a class="existingWikiWord" href="/nlab/show/background+gauge+field">background gauge field</a> induced by such a cocycle would be the corresponding <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a> to a <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle n-bundle with connection</a>. Here, since the super translation group is contractible, a <a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+lemma">Poincaré lemma</a> applies and these circle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-connections are simply given by globally defined <a class="existingWikiWord" href="/nlab/show/connection+on+an+infinity-bundle">connection</a> form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math> satisfying</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>β</mi><mo>=</mo><mi>μ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> d \beta = \mu \,. </annotation></semantics></math></div> <p>The WZW part of the GS action is then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>WZW</mi></msub><mo>:</mo><mi>ϕ</mi><mo>↦</mo><msub><mo>∫</mo> <mi>Σ</mi></msub><msup><mi>ϕ</mi> <mo>*</mo></msup><mi>β</mi></mrow><annotation encoding="application/x-tex"> S_{WZW } : \phi \mapsto \int_\Sigma \phi^* \beta </annotation></semantics></math></div> <p>(…)</p> <h2 id="properties">Properties</h2> <h3 id="KappaSymmetry">Siegel- or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>-symmetry</h3> <p>The Green-Schwarz action has an extra fermionic symmetry, on top of the genuine supersymmetry, first observed in (<a href="#Siegel83">Siegel 83</a>) for the <a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a> and in (<a href="#Siegel84">Siegel 84</a>) for the <a class="existingWikiWord" href="/nlab/show/super+1-brane+in+3d">super 1-brane in 3d</a>, and finally in (<a href="#GreenSchwarz">GreenSchwarz 84</a>) for the critical superstring in 10-dimensions. This is also called <strong><a class="existingWikiWord" href="/nlab/show/kappa-symmetry">kappa-symmetry</a></strong>. It has a natural interpretation in terms of the <a class="existingWikiWord" href="/nlab/show/supergeometry">super</a>-<a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a> of <a class="existingWikiWord" href="/nlab/show/target+space">target space</a> (<a href="#McArthur">McArthur</a>, <a href="#GKW">GKW</a>). Discussion from the point of view of the <a class="existingWikiWord" href="/nlab/show/D%27Auria-Fr%C3%A9+formulation+of+supergravity">D'Auria-Fré formulation of supergravity</a> is in (<a href="#AFFFTT98">AFFFTT 98, section 3</a>, <a href="#FreGrassi07">Fré-Grassi 07, section 2.2</a>).</p> <h3 id="BraneScan">Dimensions – the brane scan</h3> <p>The Green-Schwarz action functional of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-brane propagating on an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>-dimensional target spacetimes makes sense only for special combinations of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(p,d)</annotation></semantics></math>, for which there are suitanble <a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">super Lie algebra cocycles</a> on the super translation Lie algebra (see <a href="#DefinitionWZWTerm">above</a>).</p> <p>The corresponding table has been called the <strong>brane scan</strong> in the literature, now often called the “old brane scan”, since it has meanwhile been further completed (see below). In (<a href="#Duff87">Duff 87</a>) the “old brane scan” is displayed as follows.</p> <p><img src="https://ncatlab.org/nlab/files/DuffBraneScan.jpg" height="500" /></p> <p>In the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo><mn>10</mn></mrow><annotation encoding="application/x-tex">D = 10</annotation></semantics></math>-row we see the critical <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a> of <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> and its <a class="existingWikiWord" href="/nlab/show/electric-magnetic+duality">magnetic dual</a>, the <a class="existingWikiWord" href="/nlab/show/NS5-brane">NS5-brane</a>. The top row shows the <a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a> in <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a>.</p> <p>Moving down and left the diagonals corresponds to <em><a class="existingWikiWord" href="/nlab/show/double+dimensional+reduction">double dimensional reduction</a></em>.</p> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>The first non-empty column of the table is a reflection of the exceptional isomorphisms of the <a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a> in low dimensions and the <a class="existingWikiWord" href="/nlab/show/normed+division+algebras">normed division algebras</a>:</p> <div> <p><strong><a href="spin+group#ExceptionalIsomorphisms">exceptional</a> <a class="existingWikiWord" href="/nlab/show/spin+representation">spinors</a> and <a class="existingWikiWord" href="/nlab/show/real+numbers">real</a> <a class="existingWikiWord" href="/nlab/show/normed+division+algebras">normed division algebras</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/Lorentzian+spacetime">Lorentzian</a> <br /> <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <br /> <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>AA</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{AA}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a></th><th><a class="existingWikiWord" href="/nlab/show/normed+division+algebra">normed division algebra</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,\,</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/brane+scan">brane scan</a> entry</th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>3</mn><mo>=</mo><mn>2</mn><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">3 = 2+1</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>SL</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(2,1) \simeq SL(2,\mathbb{R})</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+1-brane+in+3d">super 1-brane in 3d</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>4</mn><mo>=</mo><mn>3</mn><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">4 = 3+1</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>SL</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(3,1) \simeq SL(2, \mathbb{C})</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+2-brane+in+4d">super 2-brane in 4d</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>6</mn><mo>=</mo><mn>5</mn><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">6 = 5+1</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>5</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo></mrow><annotation encoding="application/x-tex">Spin(5,1) \simeq</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/SL%282%2CH%29">SL(2,H)</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℍ</mi></mrow><annotation encoding="application/x-tex">\mathbb{H}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/quaternions">quaternions</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/little+string">little string</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>10</mn><mo>=</mo><mn>9</mn><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">10 = 9+1</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%289%2C1%29">Spin(9,1)</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo></mrow><annotation encoding="application/x-tex">{\simeq}</annotation></semantics></math> “<a class="existingWikiWord" href="/nlab/show/SL%282%2CO%29">SL(2,O)</a>”</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕆</mi></mrow><annotation encoding="application/x-tex">\mathbb{O}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/octonions">octonions</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/heterotic+string">heterotic</a>/<a class="existingWikiWord" href="/nlab/show/type+II+string">type II string</a></td></tr> </tbody></table> </div></div> <div class="num_remark"> <h6 id="remark_4">Remark</h6> <p>What is missing in the “old brane scan” are the <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo><mn>10</mn></mrow><annotation encoding="application/x-tex">D = 10</annotation></semantics></math> and the <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo><mn>11</mn></mrow><annotation encoding="application/x-tex">D = 11</annotation></semantics></math> (See also <a href="#BPST">BPST</a>). The reason is that the M5 corresponds to a 7-cocycle not on the ordinary <a class="existingWikiWord" href="/nlab/show/super+Poincar%C3%A9+Lie+algebra">super Poincaré Lie algebra</a>, but on its <a class="existingWikiWord" href="/nlab/show/infinity-Lie+algebra+cohomology">L-infinity algebra extension</a>, the <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a>. The completion in <a class="existingWikiWord" href="/nlab/show/super+L-infinity+algebra">super L-infinity algebra</a> theory is discussed in (<a href="#FiorenzaSatiSchreiber13">FSS 13</a>), as <em><a class="existingWikiWord" href="/schreiber/show/The+brane+bouquet">The brane bouquet</a></em>.</p> </div> <p>So (with notation as <a href="#Supercoordinates">above</a>) we have the following.</p> <div> <p>The <strong><a class="existingWikiWord" href="/nlab/show/brane">brane</a> scan</strong>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+action+functional">Green-Schwarz type</a> super <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/brane">brane</a> <a class="existingWikiWord" href="/nlab/show/sigma-models">sigma-models</a> (see at <em><a class="existingWikiWord" href="/nlab/show/table+of+branes">table of branes</a></em> for further links and see at <em><a class="existingWikiWord" href="/schreiber/show/The+brane+bouquet">The brane bouquet</a></em> for the full classification):</p> <table><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>=</mo><mi>d</mi></mover></mrow><annotation encoding="application/x-tex">\stackrel{d}{=}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">p =</annotation></semantics></math></th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>8</th><th>9</th></tr></thead><tbody><tr><td style="text-align: left;">11</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M2-brane">M2</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M5-brane">M5</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">10</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D0-brane">D0</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/F1-brane">F1</a>, <a class="existingWikiWord" href="/nlab/show/D1-brane">D1</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D2-brane">D2</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D3-brane">D3</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D4-brane">D4</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/NS5-brane">NS5</a>, <a class="existingWikiWord" href="/nlab/show/D5-brane">D5</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D6-brane">D6</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D7-brane">D7</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D8-brane">D8</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D9-brane">D9</a></td></tr> <tr><td style="text-align: left;">9</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+4-brane+in+9d">*</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">8</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+3-brane+in+8d">*</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">7</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/topological+M2-brane">M2</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi>top</mi></msub></mrow><annotation encoding="application/x-tex">{}_{top}</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">6</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/little+string">F1</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi>little</mi></msub></mrow><annotation encoding="application/x-tex">{}_{little}</annotation></semantics></math>, <a class="existingWikiWord" href="/nlab/show/self-dual+string">S1</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi>sd</mi></msub></mrow><annotation encoding="application/x-tex">{}_{sd}</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/3-brane+in+6d">S3</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">5</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+2-brane+in+5d">*</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">4</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+1-brane+in+4d">*</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+2-brane+in+4d">*</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">3</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+1-brane+in+3d">*</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> </tbody></table> <p>(The first columns follow the <a class="existingWikiWord" href="/nlab/show/exceptional+spinors+table">exceptional spinors table</a>.)</p> <p>The corresponding exceptional <a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebra">super L-∞ algebra</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">cocycles</a> (schematically, without prefactors):</p> <table><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>=</mo><mi>d</mi></mover></mrow><annotation encoding="application/x-tex">\stackrel{d}{=}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">p =</annotation></semantics></math></th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>8</th><th>9</th></tr></thead><tbody><tr><td style="text-align: left;">11</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ψ</mi> <mn>2</mn></msup><msup><mi>E</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\Psi^2 E^2</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">sIso</a>(10,1)</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ψ</mi> <mn>2</mn></msup><msup><mi>E</mi> <mn>5</mn></msup><mo>+</mo><msup><mi>Ψ</mi> <mn>2</mn></msup><msup><mi>E</mi> <mn>2</mn></msup><msub><mi>C</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">\Psi^2 E^5 + \Psi^2 E^2 C_3</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/m2brane">m2brane</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">10</td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ψ</mi> <mn>2</mn></msup><msup><mi>E</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\Psi^2 E^1</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">sIso</a>(9,1)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>B</mi> <mn>2</mn> <mn>2</mn></msubsup><mo>+</mo><msub><mi>B</mi> <mn>2</mn></msub><msup><mi>Ψ</mi> <mn>2</mn></msup><mo>+</mo><msup><mi>Ψ</mi> <mn>2</mn></msup><msup><mi>E</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">B_2^2 + B_2 \Psi^2 + \Psi^2 E^2</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/type+II+supergravity+Lie+2-algebra">StringIIA</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>⋯</mi></mrow><annotation encoding="application/x-tex">\cdots</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/type+II+supergravity+Lie+2-algebra">StringIIB</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>B</mi> <mn>2</mn> <mn>3</mn></msubsup><mo>+</mo><msubsup><mi>B</mi> <mn>2</mn> <mn>2</mn></msubsup><msup><mi>Ψ</mi> <mn>2</mn></msup><mo>+</mo><msub><mi>B</mi> <mn>2</mn></msub><msup><mi>Ψ</mi> <mn>2</mn></msup><msup><mi>E</mi> <mn>2</mn></msup><mo>+</mo><msup><mi>Ψ</mi> <mn>2</mn></msup><msup><mi>E</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">B_2^3 + B_2^2 \Psi^2 + B_2 \Psi^2 E^2 + \Psi^2 E^4</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/type+II+supergravity+Lie+2-algebra">StringIIA</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ψ</mi> <mn>2</mn></msup><msup><mi>E</mi> <mn>5</mn></msup></mrow><annotation encoding="application/x-tex">\Psi^2 E^5</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">sIso</a>(9,1)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>B</mi> <mn>2</mn> <mn>4</mn></msubsup><mo>+</mo><mi>⋯</mi><mo>+</mo><msup><mi>Ψ</mi> <mn>2</mn></msup><msup><mi>E</mi> <mn>6</mn></msup></mrow><annotation encoding="application/x-tex">B_2^4 + \cdots + \Psi^2 E^6</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/type+II+supergravity+Lie+2-algebra">StringIIA</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>⋯</mi></mrow><annotation encoding="application/x-tex">\cdots</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/type+II+supergravity+Lie+2-algebra">StringIIB</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>B</mi> <mn>2</mn> <mn>5</mn></msubsup><mo>+</mo><mi>⋯</mi><mo>+</mo><msup><mi>Ψ</mi> <mn>2</mn></msup><msup><mi>E</mi> <mn>8</mn></msup></mrow><annotation encoding="application/x-tex">B_2^5 + \cdots + \Psi^2 E^8</annotation></semantics></math> in <a class="existingWikiWord" href="/nlab/show/type+II+supergravity+Lie+2-algebra">StringIIA</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>⋯</mi></mrow><annotation encoding="application/x-tex">\cdots</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/type+II+supergravity+Lie+2-algebra">StringIIB</a></td></tr> <tr><td style="text-align: left;">9</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ψ</mi> <mn>2</mn></msup><msup><mi>E</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\Psi^2 E^4</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">sIso</a>(8,1)</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">8</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ψ</mi> <mn>2</mn></msup><msup><mi>E</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">\Psi^2 E^3</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">sIso</a>(7,1)</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">7</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ψ</mi> <mn>2</mn></msup><msup><mi>E</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\Psi^2 E^2</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">sIso</a>(6,1)</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">6</td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ψ</mi> <mn>2</mn></msup><msup><mi>E</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\Psi^2 E^1</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">sIso</a>(5,1)</td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ψ</mi> <mn>2</mn></msup><msup><mi>E</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">\Psi^2 E^3</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">sIso</a>(5,1)</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">5</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ψ</mi> <mn>2</mn></msup><msup><mi>E</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\Psi^2 E^2</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">sIso</a>(4,1)</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">4</td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ψ</mi> <mn>2</mn></msup><msup><mi>E</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\Psi^2 E^1</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">sIso</a>(3,1)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ψ</mi> <mn>2</mn></msup><msup><mi>E</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\Psi^2 E^2</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">sIso</a>(3,1)</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">3</td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ψ</mi> <mn>2</mn></msup><msup><mi>E</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\Psi^2 E^1</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">sIso</a>(2,1)</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> </tbody></table> <h3 id="the_brane_molecule">The Brane molecule</h3> <p>Furthermore, there exists a more general classification of possible supermembranes in spacetime with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> spatial dimensions and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> time dimensions, appearing in (<a href="#BlencoweDuff88">Blencowe-Duff 88</a>). In this sense, the brane scan is but the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">T=1</annotation></semantics></math> branch of the brane molecule. The objects appearing here are expected to be related to other generalizations of string theory. See <a class="existingWikiWord" href="/nlab/show/D%3D12+supergravity">D=12 supergravity</a> and <a class="existingWikiWord" href="/nlab/show/bosonic+M-theory">bosonic M-theory</a>.</p> <p><img src="/nlab/files/branemoleculepc.png" alt="The brane molecule without assuming super Poincare invariance." /></p> <p>Compare:</p> <ul> <li id="BlencoweDuff88"> <p><a class="existingWikiWord" href="/nlab/show/Miles+Blencowe">Miles Blencowe</a>, <a class="existingWikiWord" href="/nlab/show/Mike+Duff">Mike Duff</a>, <em>Supermembranes and the Signature of Space-time</em>, Nucl. Phys. B310 (1988) 387-404 (<a href="inspirehep.net/record/262142">spire:262142</a>, <a href="https://doi.org/10.1016/0550-3213(88)90155-1">10.1016/0550-3213(88)90155-1</a>, <a href="http://inspirehep.net/record/262142/files/cer-000099708.pdf">pdf</a>)</p> </li> <li> <p><em><a class="existingWikiWord" href="/nlab/show/brane+bouquet">brane bouquet</a></em></p> </li> </ul> </div> <h3 id="OnCurvedSpacetime">On curved spacetime and supergravity equations of motion</h3> <p>In the <a class="existingWikiWord" href="/nlab/show/first+order+formulation+of+gravity">first order formulation of gravity</a> a <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> configuration on a <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Cartan+connection">Cartan connection</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>∇</mo><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SuperPoincare</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex"> \nabla \colon X \to \mathbf{B} SuperPoincare(d-1,1)_{conn} </annotation></semantics></math></div> <p>hence a <a class="existingWikiWord" href="/nlab/show/principal+connection">principal connection</a> for the <a class="existingWikiWord" href="/nlab/show/super+Poincar%C3%A9+group">super Poincaré group</a> such such that at each point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math> it identifies the <a class="existingWikiWord" href="/nlab/show/tangent+space">tangent space</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>;</mo><mi>N</mi></mrow></msup><mo>=</mo><mi>𝔰𝔦𝔰𝔬</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>𝔬</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d;N} = \mathfrak{siso}(d-1,1)/\mathfrak{o}(d-1,1)</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mi>x</mi></msub><mi>X</mi><mover><mo>⟶</mo><mo>∇</mo></mover><mi>𝔰𝔦𝔰𝔬</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>⟶</mo><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>;</mo><mi>N</mi></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> T_x X \stackrel{\nabla}{\longrightarrow} \mathfrak{siso}(d-1,1) \longrightarrow \mathbb{R}^{d;N} \,. </annotation></semantics></math></div> <p>Hence given a <a class="existingWikiWord" href="/nlab/show/Lie+algebra+cocycle">Lie algebra cocycle</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi><mo>⟶</mo><mi>ℝ</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \mathfrak{g} \longrightarrow \mathbb{R}[2] </annotation></semantics></math></div> <p>as for the Green-Schwarz superstring we can <a class="existingWikiWord" href="/nlab/show/pullback+of+differential+forms">pull it back</a> along this <a class="existingWikiWord" href="/nlab/show/Cartan+connection">Cartan connection</a> to a <a class="existingWikiWord" href="/nlab/show/differential+3-form">differential 3-form</a> on <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>.</p> <p>In general this 3-form is no longer <em>closed</em>. If it is closed, then the Green-Schwarz superstring is again well defined on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo>∇</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\nabla)</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a>.</p> <p>The claim now is that requiring this 3-form still to be closed is, as a condition on the field of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math>, precisely the <a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a> of <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> (the super-<a class="existingWikiWord" href="/nlab/show/Einstein+equations">Einstein equations</a>).</p> <p>This is due to (<a href="#Nilsson81">Nilsson 81</a>, <a href="#BergshoeffSezginTownsend86">Bergshoeff-Sezgin-Townsend 86</a>) and others, see the references <a href="#ReferencesSupergravityBackgroundEquationsOfMotion">below</a>.</p> <h4 id="MembraneIn11dSuGraBackground">Membrane in 11d SuGra background</h4> <p>For the <a class="existingWikiWord" href="/nlab/show/membrane">membrane</a>(<a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a>) in a background of <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a> (<a href="#BergshoeffSezginTownsend87">Bergshoeff-Sezgin-Townsend 87</a>) find that consistency requires that (in a given <a class="existingWikiWord" href="/nlab/show/coordinate+chart">coordinate chart</a> with <a class="existingWikiWord" href="/nlab/show/super-vielbein+field">super-vielbein field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>E</mi> <mi>A</mi></msup><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msup><mi>E</mi> <mi>a</mi></msup><mo>,</mo><msup><mi>Ψ</mi> <mi>α</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E^A) = (E^a, \Psi^\alpha)</annotation></semantics></math>) the 4-form flux is of the form</p> <div class="maruku-equation" id="eq:HConstraintEquationForMembraneIn11d"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>H</mi></mtd> <mtd><mo>=</mo><mover><mi>Ψ</mi><mo>¯</mo></mover><mo>∧</mo><msup><mi>Γ</mi> <mi>ab</mi></msup><mi>Ψ</mi><mo>∧</mo><msub><mi>E</mi> <mi>a</mi></msub><mo>∧</mo><msub><mi>E</mi> <mi>b</mi></msub><mo>+</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msub><mi>C</mi> <mn>3</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>Γ</mi> <mrow><mi>a</mi><mi>b</mi><mi>α</mi><mi>β</mi></mrow></msub><msup><mi>E</mi> <mi>a</mi></msup><mo>∧</mo><msup><mi>E</mi> <mi>b</mi></msup><mo>∧</mo><msup><mi>E</mi> <mi>α</mi></msup><mo>∧</mo><msup><mi>E</mi> <mi>β</mi></msup><mo>+</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msub><mi>C</mi> <mn>3</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} H & = \overline{\Psi}\wedge\Gamma^{ab} \Psi \wedge E_a \wedge E_b + \mathbf{d}C_3 \\ & = \Gamma_{a b \alpha \beta} E^a \wedge E^b \wedge E^\alpha \wedge E^\beta + \mathbf{d}C_3 \end{aligned} </annotation></semantics></math></div> <p>where the first summand is the super-Lie algebra cocycle that classifies the <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a> and the second is the <a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a> of the <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a> proper (hence a purely bosonic differential form). In the second line we have rewritten this more manifestly in terms of the <a class="existingWikiWord" href="/nlab/show/super-vielbein">super-vielbein</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>E</mi> <mi>A</mi></msup><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msup><mi>E</mi> <mi>a</mi></msup><mo>,</mo><msup><mi>E</mi> <mi>α</mi></msup><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msup><mi>E</mi> <mi>a</mi></msup><mo>,</mo><msup><mi>Ψ</mi> <mi>α</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E^A) = (E^a, E^\alpha) = (E^a, \Psi^\alpha)</annotation></semantics></math>, this way the expression is directly analogous to that of definite 3-forms in the theory of <a class="existingWikiWord" href="/nlab/show/G%E2%82%82-manifolds">G₂-manifolds</a> (see <a href="G2+manifold#DefiniteFormsInTermsOfVielbeinFields">this example</a> for details).</p> <p>Moreover the <a class="existingWikiWord" href="/nlab/show/torsion">torsion</a> tensor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> is to have its <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>T</mi> <mi>a</mi></msup><msup><mo stretchy="false">)</mo> <mi>α</mi></msup><msub><mrow></mrow> <mi>β</mi></msub></mrow><annotation encoding="application/x-tex">(T^a)^\alpha{}_\beta</annotation></semantics></math>-component equal to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>Γ</mi> <mi>a</mi></msup><msup><mo stretchy="false">)</mo> <mi>α</mi></msup><msub><mrow></mrow> <mi>β</mi></msub></mrow><annotation encoding="application/x-tex">(\Gamma^a)^\alpha{}_\beta</annotation></semantics></math>, see at <em><a class="existingWikiWord" href="/nlab/show/torsion+constraints+in+supergravity">torsion constraints in supergravity</a></em>.</p> <p>In addition the <a class="existingWikiWord" href="/nlab/show/Bianchi+identities">Bianchi identities</a> have to hold:</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo><msup><mi>T</mi> <mi>A</mi></msup><mo>=</mo><msup><mi>E</mi> <mi>B</mi></msup><mo>∧</mo><msub><mi>R</mi> <mi>B</mi></msub><msup><mrow></mrow> <mi>A</mi></msup></mrow><annotation encoding="application/x-tex">\nabla T^A = E^B \wedge R_{B}{}^{A}</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo><mi>H</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\nabla H = 0</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/covariant+derivative">covariant constancy</a>).</p> </li> </ul> <p>All this is implied by the <a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a> of <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a>.</p> <p>Notice that in view of the above analogy to <a class="existingWikiWord" href="/nlab/show/G%E2%82%82-structure">G₂-structure</a>, the covariant constancy condition is precisely the analog of <a class="existingWikiWord" href="/nlab/show/G%E2%82%82-manifold">G₂-manifold</a> structure.</p> <p>Discussion of this in the somewhat more streamlined <a class="existingWikiWord" href="/nlab/show/D%27Auria-Fr%C3%A9+formulation+of+supergravity">D'Auria-Fré formulation of supergravity</a> is in (<a href="#AFFFTT98">AFFFTT 98, section 3.1</a>).</p> <h4 id="heterotic_string">Heterotic string</h4> <p>Discussion that for the GS-version of the <a class="existingWikiWord" href="/nlab/show/heterotic+string">heterotic string</a> consistency of the background is equivalent to the equations of motion of <a class="existingWikiWord" href="/nlab/show/heterotic+supergravity">heterotic supergravity</a> is in (<a href="#ShapiroTaylor87">Shapiro-Taylor 87</a>).</p> <p>Discussion with the hetetoric gauge field included is in (<a href="#AtickDharRatra86">Atick-Dhar-Ratra 86</a>).</p> <h4 id="type_ii_string">Type II string</h4> <p>Discussion for the GS-version of the <a class="existingWikiWord" href="/nlab/show/type+II+superstring">type II superstring</a> in <a class="existingWikiWord" href="/nlab/show/type+II+supergravity">type II supergravity</a>-backgrounds is in (<a href="#GHMNT85">GHMNT 85</a>), and for the <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a> in type II in (<a href="#CGNSW97">CGNSW 97</a>).</p> <h3 id="ConservedCurrents">Conserved currents</h3> <p>The super-WZW term of the GS action functionals is invariant under <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a> only up to a <a class="existingWikiWord" href="/nlab/show/divergence">divergence</a>. Hence the <a class="existingWikiWord" href="/nlab/show/Noether+theorem">Noether theorem</a> in its generality for “weak” symmetries applies and gives that the <a class="existingWikiWord" href="/nlab/show/conserved+currents">conserved currents</a> receive an extra contribution from this divergence term. The resulting algebra is a central extension of the given <a class="existingWikiWord" href="/nlab/show/super+translation+Lie+algebra">super translation Lie algebra</a>, extending to the famous <a href="super+Poincare+Lie+algebra#PolyvectorExtensions">polyvector extensions</a> “by brane charges” of the <a class="existingWikiWord" href="/nlab/show/super+Poincar%C3%A9+Lie+algebra">super Poincaré Lie algebra</a> (<a href="#AGIT89">AGIT 89</a>).</p> <h3 id="AsPartOfTheAdSCFTCorrespodence">As part of the AdS-CFT correspondence</h3> <p>By the above discussion, Green-Schwarz super <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-branes are consistent on <a class="existingWikiWord" href="/nlab/show/superspacetimes">superspacetimes</a> that satisfy the respective higher <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> <a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a>. These turn out to have solutions which exhibit <a class="existingWikiWord" href="/nlab/show/black+branes">black branes</a> in essentially just the combinations of dimensions and supersymmetries that the original Green-Schwarz sigma-models exist in, hence they look precisely like the <a class="existingWikiWord" href="/nlab/show/non-perturbative+quantum+field+theory">non-perturbative</a> avatars of whatever these <a class="existingWikiWord" href="/nlab/show/sigma+models">sigma models</a> give the <a class="existingWikiWord" href="/nlab/show/perturbation+theory">perturbation theory</a> of by <a class="existingWikiWord" href="/nlab/show/second+quantization">second quantization</a>. (See at <a class="existingWikiWord" href="/nlab/show/black+holes+in+string+theory">black holes in string theory</a> for more on this correspondence between branes in string perturbation theory and black branes in supergravity.)</p> <p>Moreover, the near-horizon geometries of these <a class="existingWikiWord" href="/nlab/show/black+branes">black branes</a> are always <a class="existingWikiWord" href="/nlab/show/anti+de+Sitter+spacetime">anti de Sitter spacetime</a> times orthogonal directions.</p> <p>Therefore it is natural to consider the perturbation of the Green-Schwarz sigma-models around their asymptotic embeddings into AdS spaces, hence effectively the perturbation theory of the degrees of freedom at those naked singularity at which the corresponding black brane sits.</p> <p>After <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphism</a> <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixing</a> one finds that the resulting field theories now on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-brane <a class="existingWikiWord" href="/nlab/show/worldvolumes">worldvolumes</a> are precisely the <a class="existingWikiWord" href="/nlab/show/superconformal+field+theories">superconformal field theories</a> for all the <a href="supersymmetry#ClassificationSuperconformalInDimgt2">allowed superconformal supersymmetries</a> (see also at <em><a class="existingWikiWord" href="/nlab/show/singleton+representation">singleton representation</a></em>):</p> <table><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math></th><th><a class="existingWikiWord" href="/nlab/show/superconformal+super+Lie+algebra">superconformal super Lie algebra</a></th><th><a class="existingWikiWord" href="/nlab/show/R-symmetry">R-symmetry</a></th><th><a class="existingWikiWord" href="/nlab/show/brane">brane</a> <a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a> theory</th></tr></thead><tbody><tr><td style="text-align: left;">3</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">2k+1</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo><mo>≃</mo></mrow><annotation encoding="application/x-tex">B(k,2) \simeq </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/osp">osp</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">/</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2k+1/4)</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(2k+1)</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">3</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>k</mi></mrow><annotation encoding="application/x-tex">2k</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo><mo>≃</mo></mrow><annotation encoding="application/x-tex">D(k,2)\simeq </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/osp">osp</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>k</mi><mo stretchy="false">/</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2k/4)</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>2</mn><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(2k)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a></td></tr> <tr><td style="text-align: left;">4</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">k+1</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">(</mo><mn>3</mn><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>𝔰𝔩</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">/</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A(3,k)\simeq \mathfrak{sl}(4/k+1)</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(k+1)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D3-brane">D3-brane</a></td></tr> <tr><td style="text-align: left;">5</td><td style="text-align: left;">1</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(4)</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(3)</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">6</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mn>4</mn><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mo>≃</mo></mrow><annotation encoding="application/x-tex">D(4,k) \simeq</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/osp">osp</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>8</mn><mo stretchy="false">/</mo><mn>2</mn><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(8/2k)</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sp</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sp(k)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a></td></tr> </tbody></table> <p>This is effectively the <a class="existingWikiWord" href="/nlab/show/AdS-CFT+correspondence">AdS-CFT correspondence</a>.</p> <p>Detailed discussion of the above steps is in (<a href="#AFFFTT98">AFFFTT 98</a>, <a href="#PastiSorokinTonin99">Pasti-Sorokin-Tonin 99</a>). The resulting “<a class="existingWikiWord" href="/nlab/show/superconformal+algebra">super-conformal</a> brane scan” is discussed in <a href="#BlencoweDuff88">Blencowe & Duff 1988</a> with further developments in <a href="#Duff09">Duff 2009</a>, <a href="#Duff21">Duff 2021</a>.</p> <h3 id="quantization">Quantization</h3> <p>The <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> of the Green-Schwarz super <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-brane <a class="existingWikiWord" href="/nlab/show/sigma+models">sigma models</a> is discussed in the literature in terms of <a class="existingWikiWord" href="/nlab/show/light-cone+gauge+quantization">light-cone gauge quantization</a>.</p> <p>This is actually how the Green-Schwarz superstring was first introduced in (<a href="#GreenSchwarz81">Green-Schwarz 81</a>, <a href="#GreenSchwarz82">Green-Schwarz 82</a>) before its <a class="existingWikiWord" href="/nlab/show/general+covariance">generally covariant</a> formulation was found in (<a href="#GreenSchwarz84">Green-Schwarz 84</a>). A textbook account of this is in (<a href="#GreenSchwarzWitten">Green-Schwarz-Witten, section 5</a>).</p> <p>While, by the <a class="existingWikiWord" href="/nlab/show/brane+scan">brane scan</a> discussed above, the action functional for the Green-Schwarz superstring exists for target <a class="existingWikiWord" href="/nlab/show/super+Minkowski+spacetimes">super Minkowski spacetimes</a> of dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">d = 3</annotation></semantics></math>, 4, 6, and 10, its <a class="existingWikiWord" href="/nlab/show/light-cone+gauge+quantization">light-cone gauge quantization</a> produces a <a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a> for the spacetime <a class="existingWikiWord" href="/nlab/show/Lorentz+group">Lorentz group</a> symmetry in dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">d = 4</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><mn>6</mn></mrow><annotation encoding="application/x-tex">d = 6</annotation></semantics></math>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><mn>10</mn></mrow><annotation encoding="application/x-tex">d = 10</annotation></semantics></math> the anomaly disappears and the thus quantized Green-Schwarz string becomes equivalent to the quantum <a class="existingWikiWord" href="/nlab/show/NSR+string">NSR string</a>, hence to “the” critical string (of <a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a>, <a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a>).</p> <p>Curiously, the light-cone gauge quantization of the GS-string also does wor however for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">d = 3</annotation></semantics></math>, see at <em><a class="existingWikiWord" href="/nlab/show/super+1-brane+in+3d">super 1-brane in 3d</a></em> for more on this.</p> <p>(…)</p> <h2 id="related_entries">Related entries</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/brane+scan">brane scan</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/brane+bouquet">brane bouquet</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a>, <a class="existingWikiWord" href="/nlab/show/brane">brane</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/relativistic+particle">relativistic particle</a>, <a class="existingWikiWord" href="/nlab/show/spinning+particle">spinning particle</a>, <a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string">string</a>, <a class="existingWikiWord" href="/nlab/show/spinning+string">spinning string</a>, <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/membrane">membrane</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+fundamental+super+p-branes">geometry of physics – fundamental super p-branes</a></p> </li> </ul> <h2 id="references">References</h2> <h3 id="SuperStringAsAGSSigmaModel">Super-string as a GS-sigma model</h3> <p>A precursor to the actual Green-Schwarz action functional is</p> <ul> <li id="GreenSchwarz81"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Green">Michael Green</a>, <a class="existingWikiWord" href="/nlab/show/John+Schwarz">John Schwarz</a>, <em>Supersymmetrical Dual String Theory</em>, Nucl. Phys. B 181 (1981) 502;</p> </li> <li id="GreenSchwarz82"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Green">Michael Green</a>, <a class="existingWikiWord" href="/nlab/show/John+Schwarz">John Schwarz</a>, <em>Supersymmetrical Dual String Theory. 2. Vertices and Trees</em>, Nucl. Phys. B 198 (1982) 252.</p> </li> </ul> <p>which presented a <a class="existingWikiWord" href="/nlab/show/light-cone+gauge+quantization">light-cone gauge quantization</a> of superstring with manifest <a class="existingWikiWord" href="/nlab/show/target+space">target</a> <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a>.</p> <p>The observation that this has a <a class="existingWikiWord" href="/nlab/show/general+covariance">generally covariant</a> formulation lead to what is now called the Green-Schwarz action functional proper, for the superstring:</p> <ul> <li id="GreenSchwarz84"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Green">Michael Green</a>, <a class="existingWikiWord" href="/nlab/show/John+Schwarz">John Schwarz</a>, <em>Covariant description of superstrings</em>, Phys. Lett. B136 (1984), 367–370 (<a href="http://inspirehep.net/record/193596">spire:193596</a>, <a href="https://doi.org/10.1016/0370-2693(84)92021-5">arXiv;10.1016/0370-2693(84)92021-5</a>)</p> </li> <li id="GreenSchwarz84b"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Green">Michael Green</a>, <a class="existingWikiWord" href="/nlab/show/John+Schwarz">John Schwarz</a>, <em>Properties of the Covariant Formulation of Superstring Theories</em>, Nucl. Phys. B 243 (1984) 285 (<a href="http://inspirehep.net/record/196623">spire:196623</a>, <a href="https://doi.org/10.1016/0550-3213(84)90030-0">doi:10.1016/0550-3213(84)90030-0</a>)</p> </li> </ul> <p>See also the historical comments in</p> <ul> <li id="Schwarz16"><a class="existingWikiWord" href="/nlab/show/John+Schwarz">John Schwarz</a>, slides 24-25 of <em>String Theory in the Twentieth Century</em>, talk at <a href="http://ymsc.tsinghua.edu.cn:8090/strings/">Strings 2016</a> (<a href="http://ymsc.tsinghua.edu.cn:8090/strings/slides/8.1/Schwarz.pdf">pdf</a>)</li> </ul> <p>Early discussion of coupling the GS superstring to general <a class="existingWikiWord" href="/nlab/show/background+fields">background fields</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">D=4</annotation></semantics></math>:</p> <ul> <li id="Gates90"><a class="existingWikiWord" href="/nlab/show/S.+James+Gates+Jr.">S. James Gates Jr.</a>: <em>Progress Toward Covariant Formulation of All <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">D=4</annotation></semantics></math> GS-type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-model Actons</em>, in: <em>Superstrings and Particle Theory</em>, Proceedings of the Conference in Tuscaloosa 8–11 November 1989, World Scientific (1990) [<a href="https://doi.org/10.1142/1051">doi:10.1142/1051</a>, <a class="existingWikiWord" href="/nlab/files/Gates-D4GSModels.pdf" title="pdf">pdf</a>]</li> </ul> <p>A standard textbook reference for the GS superstring is</p> <ul> <li id="GreenSchwarzWitten"><a class="existingWikiWord" href="/nlab/show/Michael+Green">Michael Green</a>, <a class="existingWikiWord" href="/nlab/show/John+Schwarz">John Schwarz</a>, <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, volume 1, section 5 of <em>Superstring theory</em>, 3 vols. Cambridge Monographs on Mathematical Physics</li> </ul> <p>and a brief paragraph in Volume II, section 10.2, page 983 of</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Eric+D%27Hoker">Eric D'Hoker</a>, <em>String theory – lecture 10: Supersymmetry and supergravity</em> , in part 3 of</p> <p><a class="existingWikiWord" href="/nlab/show/Pierre+Deligne">Pierre Deligne</a>, <a class="existingWikiWord" href="/nlab/show/Pavel+Etingof">Pavel Etingof</a>, <a class="existingWikiWord" href="/nlab/show/Dan+Freed">Dan Freed</a>, L. Jeffrey,</p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/David+Kazhdan">David Kazhdan</a>, <a class="existingWikiWord" href="/nlab/show/John+Morgan">John Morgan</a>, D.R. Morrison and <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, eds. <em><a class="existingWikiWord" href="/nlab/show/Quantum+Fields+and+Strings">Quantum Fields and Strings</a>, A course for mathematicians</em>, 2 vols. Amer. Math. Soc. Providence 1999. (<a href="http://www.math.ias.edu/qft">web version</a>)</p> <p>Textbook discussion of the Green-Schwarz version of the <a class="existingWikiWord" href="/nlab/show/heterotic+string">heterotic string</a> is in</p> <ul> <li id="CastellaniDAuriaFre91"><a class="existingWikiWord" href="/nlab/show/Leonardo+Castellani">Leonardo Castellani</a>, <a class="existingWikiWord" href="/nlab/show/Riccardo+D%27Auria">Riccardo D'Auria</a>, <a class="existingWikiWord" href="/nlab/show/Pietro+Fr%C3%A9">Pietro Fré</a>, vol 3, section VI.9.7 of <em><a class="existingWikiWord" href="/nlab/show/Supergravity+and+Superstrings+-+A+Geometric+Perspective">Supergravity and Superstrings - A Geometric Perspective</a></em>, World Scientific (1991)</li> </ul> <p>Comprehensive review:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Joan+Sim%C3%B3n">Joan Simón</a>, <em>Brane Effective Actions, Kappa-Symmetry and Applications</em>, Living Reviews in Relativity <strong>15</strong> 3 (2012) [<a href="http://arxiv.org/abs/1110.2422">arXiv:1110.2422</a>, <a href="https://doi.org/10.12942/lrr-2012-3">doi:10.12942/lrr-2012-3</a>]</li> </ul> <p>Quantization of the Green-Schwarz string using <a class="existingWikiWord" href="/nlab/show/pure+spinors">pure spinors</a> is discussed in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Pietro+A.+Grassi">Pietro A. Grassi</a>, <a class="existingWikiWord" href="/nlab/show/Giuseppe+Policastro">Giuseppe Policastro</a>, <a class="existingWikiWord" href="/nlab/show/Peter+van+Nieuwenhuizen">Peter van Nieuwenhuizen</a>, <em>An Introduction to the Covariant Quantization of Superstrings</em>, Class. Quant. Grav. <strong>20</strong> (2003) 395-410 [<a href="https://arxiv.org/abs/hep-th/0302147">arXiv:hep-th/0302147</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pietro+A.+Grassi">Pietro A. Grassi</a>, Y. Oz, <em>Non-Critical Covariant Superstrings</em> [<a href="https://arxiv.org/abs/hep-th/0507168">arXiv:hep-th/0507168</a>]</p> </li> </ul> <p>Review in the <a class="existingWikiWord" href="/nlab/show/superembedding+approach">superembedding approach</a></p> <ul> <li id="Sorokin99"><a class="existingWikiWord" href="/nlab/show/Dmitri+Sorokin">Dmitri Sorokin</a>, <em>Superbranes and Superembeddings</em>, Phys. Rept. <strong>329</strong> (2000) 1-101 [<a href="http://arxiv.org/abs/hep-th/9906142">arXiv:hep-th/9906142</a>]</li> </ul> <p>The observation that the Green-Schwarz action functional is an example of a <a class="existingWikiWord" href="/nlab/show/WZW-model">WZW-model</a> on <a class="existingWikiWord" href="/nlab/show/super-Minkowski+spacetime">super-Minkowski spacetime</a> is due to</p> <ul> <li id="HenneauxMezincescu85"><a class="existingWikiWord" href="/nlab/show/Marc+Henneaux">Marc Henneaux</a>, Luca Mezincescu, <em>A <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-Model Interpretation of Green-Schwarz Covariant Superstring Action</em>, Phys. Lett. B <strong>152</strong> (1985) 340 [<a href="https://doi.org/10.1016/0370-2693(85)90507-6">doi:10.1016/0370-2693(85)90507-6</a>, <a href="https://inspirehep.net/literature/15922">spire:15922</a>]</li> </ul> <p>For more references on this WZW perspective see <a href="#ReferencesWZWTerm">below</a>.</p> <p>For references on curved backgrounds see <a href="#ReferencesSupergravityBackgroundEquationsOfMotion">below</a>.</p> <div> <h3 id="SuperMembraneAsSigmaModelReferences">Super-membrane/M2-brane as a sigma model</h3> <p>The <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+sigma-model">Green-Schwarz sigma-model</a>-type formulation of the <a class="existingWikiWord" href="/nlab/show/super-membrane">super-membrane</a> in 11d (as in the <a class="existingWikiWord" href="/nlab/show/brane+scan">brane scan</a> and in contrast to the <a class="existingWikiWord" href="/nlab/show/black+brane">black brane</a>-solutions of <a class="existingWikiWord" href="/nlab/show/11d+supergravity">11d supergravity</a>) first appears in:</p> <ul> <li id="BergshoeffSezginTownsend87"><a class="existingWikiWord" href="/nlab/show/Eric+Bergshoeff">Eric Bergshoeff</a>, <a class="existingWikiWord" href="/nlab/show/Ergin+Sezgin">Ergin Sezgin</a>, and <a class="existingWikiWord" href="/nlab/show/Paul+Townsend">Paul Townsend</a>, <em>Supermembranes and eleven-dimensional supergravity</em>, Phys. Lett. B <strong>189</strong> (1987) 75-78 [<a href="https://doi.org/10.1016/0370-2693(87)91272-X">doi:10.1016/0370-2693(87)91272-X</a>, <a href="http://inspirehep.net/record/248230/">spire:248230</a>]</li> </ul> <p>Via the <a class="existingWikiWord" href="/nlab/show/superembedding+approach">superembedding approach</a> the <a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a> were obtained in</p> <ul> <li id="BPSTV95"><a class="existingWikiWord" href="/nlab/show/Igor+Bandos">Igor Bandos</a>, <a class="existingWikiWord" href="/nlab/show/Paolo+Pasti">Paolo Pasti</a>, <a class="existingWikiWord" href="/nlab/show/Dmitri+Sorokin">Dmitri Sorokin</a>, <a class="existingWikiWord" href="/nlab/show/Mario+Tonin">Mario Tonin</a>, <a class="existingWikiWord" href="/nlab/show/Dmitry+Volkov">Dmitry Volkov</a>, Chapter 3 of <em>Superstrings and supermembranes in the doubly supersymmetric geometrical approach</em>, Nucl. Phys. B446:79-118, 1995 (<a href="https://arxiv.org/abs/hep-th/9501113">arXiv:hep-th/9501113</a>)</li> </ul> <p>and the <a class="existingWikiWord" href="/nlab/show/Lagrangian+density">Lagrangian density</a> in:</p> <ul> <li id="HoweSezgin05"><a class="existingWikiWord" href="/nlab/show/Paul+Howe">Paul Howe</a>, <a class="existingWikiWord" href="/nlab/show/Ergin+Sezgin">Ergin Sezgin</a>, <em>The supermembrane revisited</em>, Class. Quant. Grav. <strong>22</strong> (2005) 2167-2200 [<a href="https://arxiv.org/abs/hep-th/0412245">arXiv:hep-th/0412245</a>, <a href="https://doi.org/10.1088/0264-9381/22/11/017">doi:10.1088/0264-9381/22/11/017</a>]</li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/double+dimensional+reduction">double dimensional reduction</a> of the M2-brane to the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+superstring">Green-Schwarz superstring</a> was observed in</p> <ul> <li id="DuffHoweInamiStelle87"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Duff">Michael Duff</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Howe">Paul Howe</a>, T. Inami, <a class="existingWikiWord" href="/nlab/show/Kellogg+Stelle">Kellogg Stelle</a>, <em>Superstrings in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo><mn>10</mn></mrow><annotation encoding="application/x-tex">D=10</annotation></semantics></math> from Supermembranes in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo><mn>11</mn></mrow><annotation encoding="application/x-tex">D=11</annotation></semantics></math></em>, Phys. Lett. B <strong>191</strong> (1987) 70 [<a href="https://doi.org/10.1016/0370-2693(87)91323-2">doi:10.1016/0370-2693(87)91323-2</a>]</p> <p>also in: <a class="existingWikiWord" href="/nlab/show/Michael+Duff">Michael Duff</a> (ed.): <em><a class="existingWikiWord" href="/nlab/show/The+World+in+Eleven+Dimensions">The World in Eleven Dimensions</a></em> 205-206 (1987) [<a href="http://inspirehep.net/record/245249">spire:245249</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Paul+Townsend">Paul Townsend</a>, <em>The eleven-dimensional supermembrane revisited</em>, Phys. Lett. B <strong>350</strong> (1995) 184-187 [<a href="http://arxiv.org/abs/hep-th/9501068">arXiv:hep-th/9501068</a>, <a href="https://doi.org/10.1016/0370-2693(95)00397-4">doi:10.1016/0370-2693(95)00397-4</a>]</p> </li> </ul> <p>around the time when <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a> became accepted due to</p> <ul> <li id="Witten95"><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em><a class="existingWikiWord" href="/nlab/show/String+Theory+Dynamics+In+Various+Dimensions">String Theory Dynamics In Various Dimensions</a></em> (<a href="http://arxiv.org/abs/hep-th/9503124">arXiv:hep-th/9503124</a>)</li> </ul> <p>See also</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Igor+Bandos">Igor Bandos</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Townsend">Paul Townsend</a>, <em>SDiff Gauge Theory and the M2 Condensate</em> (<a href="http://arxiv.org/abs/0808.1583">arXiv:0808.1583</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Maria+P.+Garcia+del+Moral">Maria P. Garcia del Moral</a>, C. Las Heras, P. Leon, J. M. Pena, <a class="existingWikiWord" href="/nlab/show/Alvaro+Restuccia">Alvaro Restuccia</a>, <em>Fluxes, Twisted tori, Monodromy and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math> Supermembranes</em>, J. High Energ. Phys. <strong>2020</strong> 97 (2020) &lbrack;<a href="https://arxiv.org/abs/2005.06397">arXiv:2005.06397</a>, <a href="https://doi.org/10.1007/JHEP09(2020)097">doi:10.1007/JHEP09(2020)097</a>&rbrack;</p> </li> </ul> <p>Discussion from the point of view of <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+action+functional">Green-Schwarz action functional</a>-<a class="existingWikiWord" href="/schreiber/show/%E2%88%9E-Wess-Zumino-Witten+theory">∞-Wess-Zumino-Witten theory</a>:</p> <ul> <li id="FSS2013"><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/The+brane+bouquet">Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields</a></em> (2013)</li> </ul> <p>On possible structures in M2-brane dynamics and <a class="existingWikiWord" href="/nlab/show/M2-M5-brane+bound+states">M2-M5-brane bound states</a> which could be <a class="existingWikiWord" href="/nlab/show/M-theory">M-theory</a>-lifts of the familiar <a class="existingWikiWord" href="/nlab/show/integrable+system">integrability</a> of the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+superstring">Green-Schwarz superstring</a> on <a class="existingWikiWord" href="/nlab/show/AdS+spacetime"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>AdS</mi> <mn>5</mn></msub> </mrow> <annotation encoding="application/x-tex">AdS_5</annotation> </semantics> </math></a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>×</mo></mrow><annotation encoding="application/x-tex">\times</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/5-sphere"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msup><mi>S</mi> <mn>5</mn></msup> </mrow> <annotation encoding="application/x-tex">S^5</annotation> </semantics> </math></a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kirill+Gubarev">Kirill Gubarev</a>, <a class="existingWikiWord" href="/nlab/show/Edvard+Musaev">Edvard Musaev</a>, <em>Integrability structures in string theory</em> &lbrack;<a href="https://arxiv.org/abs/2301.06486">arXiv:2301.06486</a>&rbrack;</li> </ul> </div> <h3 id="ReferencesForDBranes">D-branes as GS-sigma models</h3> <p>Green-Schwarz action functionals for the <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a> (including the <a class="existingWikiWord" href="/nlab/show/DBI+action">DBI action</a>):</p> <ul> <li id="CGNSW96"> <p><a class="existingWikiWord" href="/nlab/show/Martin+Cederwall">Martin Cederwall</a>, Alexander von Gussich, <a class="existingWikiWord" href="/nlab/show/Bengt+Nilsson">Bengt Nilsson</a>, Per Sundell, Anders Westerberg, <em>The Dirichlet Super-p-Branes in Ten-Dimensional Type IIA and IIB Supergravity</em>, Nucl.Phys. B490 (1997) 179-201 (<a href="http://arxiv.org/abs/hep-th/9611159">arXiv:hep-th/9611159</a>)</p> </li> <li id="APPS97b"> <p><a class="existingWikiWord" href="/nlab/show/Mina+Aganagic">Mina Aganagic</a>, <a class="existingWikiWord" href="/nlab/show/Jaemo+Park">Jaemo Park</a>, Costin Popescu, <a class="existingWikiWord" href="/nlab/show/John+Schwarz">John Schwarz</a>, <em>Dual D-Brane Actions</em>, Nucl. Phys. B496 (1997) 215-230 (<a href="https://arxiv.org/abs/hep-th/9702133">arXiv:hep-th/9702133</a>)</p> </li> <li> <p><a href="#CAIB99">CAIB 99</a></p> </li> <li> <p><a href="#Sakaguchi00">Sakaguchi 00</a></p> </li> </ul> <h3 id="dualities">Dualities</h3> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/T-duality">T-duality</a> for the Green-Schwarz string is in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Mirjam+Cvetic">Mirjam Cvetic</a>, H. Lu, <a class="existingWikiWord" href="/nlab/show/Christopher+Pope">Christopher Pope</a>, <a class="existingWikiWord" href="/nlab/show/Kellogg+Stelle">Kellogg Stelle</a>, <em>T-Duality in the Green-Schwarz Formalism, and the Massless/Massive IIA Duality Map</em>, Nucl.Phys.B573:149-176,2000 (<a href="https://arxiv.org/abs/hep-th/9907202">arXiv:hep-th/9907202</a>)</p> </li> <li> <p>Bogdan Kulik, Radu Roiban, <em>T-duality of the Green-Schwarz superstring</em>, JHEP 0209 (2002) 007 (<a href="https://arxiv.org/abs/hep-th/0012010">arXiv:hep-th/0012010</a>)</p> </li> <li id="BandosJulia03"> <p><a class="existingWikiWord" href="/nlab/show/Igor+Bandos">Igor Bandos</a>, <a class="existingWikiWord" href="/nlab/show/Bernard+Julia">Bernard Julia</a>, <em>Superfield T-duality rules</em>, JHEP 0308 (2003) 032 (<a href="https://arxiv.org/abs/hep-th/0303075">arXiv:hep-th/0303075</a>)</p> <p>reviewed in <em>Superfield T-duality rules in ten dimensions with one isometry</em> (<a href="https://arxiv.org/abs/hep-th/0312299">arXiv:hep-th/0312299</a>)</p> </li> </ul> <h3 id="ReferencesWZWTerm">WZW terms, super Lie algebra cohomology and the brane scan</h3> <p>The WZW nature of the second term in the GS action, recognized in (<a href="#HenneauxMezincescu85">Henneaux-Mezincescu 85</a>) is discussed in</p> <ul> <li id="Rabin87"> <p><a class="existingWikiWord" href="/nlab/show/Jeffrey+Rabin">Jeffrey Rabin</a>: <em>Supermanifold Cohomology and the Wess-Zumino Term of the Covariant Superstring Action</em>, Commun Math. Phys. <strong>108</strong> (1987) 375-389 [<a href="https://projecteuclid.org/euclid.cmp/1104116532">euclid:cmp/1104116532</a>]</p> </li> <li> <p>B. Milewski, <em>Superstrings from sigma models</em>, Classical and Quantum Gravity, Volume 4, Number 3 (1987)</p> </li> <li> <p>A.P. Isaev, E.A. Ivanov, <em>On Sigma Model Formulation of Green-Schwarz Superstring</em>, Mod.Phys.Lett. A4 (1989) 351-359 (<a href="http://inspirehep.net/record/266793">spire:266793</a>, <a href="https://www.worldscientific.com/doi/abs/10.1142/S0217732389000423">doi:10.1142/S0217732389000423</a>)</p> </li> <li id="AzcarragaTownsend89"> <p><a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+de+Azc%C3%A1rraga">José de Azcárraga</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Townsend">Paul Townsend</a>: <em>Superspace geometry and the classification of supersymmetric extended objects</em>, Physical Review Letters <strong>62</strong> 22 (1989) [<a href="https://doi.org/10.1103/PhysRevLett.62.2579">doi:10.1103/PhysRevLett.62.2579</a>, <a href="http://inspirehep.net/record/284635">spire:284635</a>]</p> </li> </ul> <p>and with its <a class="existingWikiWord" href="/nlab/show/infinity-Lie+theory">Lie theoretic</a> meaning made fully explicit (in “FDA” language) in</p> <ul> <li id="AzcarragaIzquierdo95"><a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+de+Azc%C3%A1rraga">José de Azcárraga</a>, <a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+Izquierdo">José Izquierdo</a>, chapter 8 of <em><a class="existingWikiWord" href="/nlab/show/Lie+Groups%2C+Lie+Algebras%2C+Cohomology+and+Some+Applications+in+Physics">Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics</a></em>, Cambridge monographs of Mathematical Physics (1995) [<a href="https://doi.org/10.1017/CBO9780511599897">doi:10.1017/CBO9780511599897</a>]</li> </ul> <p>The original “brane scan” classification of GS action functionals by WZW terms:</p> <ul> <li id="AETW87">Anna Achúcarro, <a class="existingWikiWord" href="/nlab/show/Jonathan+Evans">Jonathan Evans</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Townsend">Paul Townsend</a>, <a class="existingWikiWord" href="/nlab/show/David+Wiltshire">David Wiltshire</a>, <em>Super <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-Branes</em>, Phys. Lett. B <strong>198</strong> (1987) 441 [<a href="http://inspirehep.net/record/22286">spire:22286</a>, <a href="https://doi.org/10.1016/0370-2693(87)90896-3">doi:10.1016/0370-2693(87)90896-3</a>, <a href="https://ir.canterbury.ac.nz/xmlui/bitstream/handle/10092/9041/12616571_pbrane.pdf?sequence=1&isAllowed=y">pdf</a>, <a class="existingWikiWord" href="/nlab/files/SuperpBranes.pdf" title="pdf">pdf</a>]</li> </ul> <p>A complete rigorous classification of all the relevant cocycles on the <a class="existingWikiWord" href="/nlab/show/super+Poincar%C3%A9+Lie+algebra">super Poincaré Lie algebra</a> was given in</p> <ul> <li id="Brandt12-13"> <p><a class="existingWikiWord" href="/nlab/show/Friedemann+Brandt">Friedemann Brandt</a>, <em>Supersymmetry algebra cohomology</em></p> <p><em>I: Definition and general structure</em> J. Math. Phys.51:122302, 2010, (<a href="http://arxiv.org/abs/0911.2118">arXiv:0911.2118</a>)</p> <p><em>II: Primitive elements in 2 and 3 dimensions</em>, J. Math. Phys. 51 (2010) 112303 (<a href="http://arxiv.org/abs/1004.2978">arXiv:1004.2978</a>)</p> <p><em>III: Primitive elements in four and five dimensions</em>, J. Math. Phys. 52:052301, 2011 (<a href="http://arxiv.org/abs/1005.2102">arXiv:1005.2102</a>)</p> <p><em>IV: Primitive elements in all dimensions from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">D=4</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo><mn>11</mn></mrow><annotation encoding="application/x-tex">D=11</annotation></semantics></math></em>, J. Math. Phys. 54, 052302 (2013) (<a href="http://arxiv.org/abs/1303.6211">arXiv:1303.6211</a>)</p> </li> </ul> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><mn>11</mn></mrow><annotation encoding="application/x-tex">d = 11</annotation></semantics></math> the relevant <a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a> <a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">cocycles</a> have also been discussed (but not related to the Green-Schwarz action functional) in</p> <ul> <li id="DAuriaFre82"><a class="existingWikiWord" href="/nlab/show/Riccardo+D%27Auria">Riccardo D'Auria</a>, <a class="existingWikiWord" href="/nlab/show/Pietro+Fr%C3%A9">Pietro Fré</a>, <em><a class="existingWikiWord" href="/nlab/files/GeometricSupergravity.pdf" title="Geometric Supergravity in D=11 and its hidden supergroup">Geometric Supergravity in D=11 and its hidden supergroup</a></em>, Nuclear Physics B201 (1982)</li> </ul> <p>A review is in</p> <ul> <li id="Duff87"><a class="existingWikiWord" href="/nlab/show/Michael+Duff">Michael Duff</a>, <em>Supermembranes: the first fifteen weeks</em>, Class. Quant. Grav. <strong>5</strong> (1988) 189 [<a href="https://doi.org/10.1088/0264-9381/5/1/023">doi:10.1088/0264-9381/5/1/023</a>, <a href="https://inspirehep.net/record/248034">spire:248034</a>]</li> </ul> <p>from which the above table is taken.</p> <p>Systematic review and discussion of the 3- and 4-cocycles in the old <a class="existingWikiWord" href="/nlab/show/brane+scan">brane scan</a> via the relation between <a class="existingWikiWord" href="/nlab/show/division+algebras+and+supersymmetry">division algebras and supersymmetry</a> is in</p> <ul> <li id="BaezHuerta10"><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, <a class="existingWikiWord" href="/nlab/show/John+Huerta">John Huerta</a>, <em>Division algebras and supersymmetry II</em>, Adv. Math. Theor. Phys. 15 (2011), 1373-1410 (<a href="http://arxiv.org/abs/1003.3436">arXiv:1003.34360</a>)</li> </ul> <p>See also</p> <ul> <li>I. Bars, C. Deliduman and D. Minic, Phys. Rev D59 (1999) 125004; Phys. Lett. B457 (1999) 275. (<a href="http://arxiv.org/abs/hep-th/9812161">arXiv:hep-th/9812161</a>)</li> </ul> <p>More along these lines is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Michael+Duff">Michael Duff</a>, S. Ferrara, <em>Four curious supergravities</em> (<a href="http://arxiv.org/abs/1010.3173">arXiv</a>)</li> </ul> <p>The Green-Schwarz-type action for the <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> was found in</p> <ul> <li id="BLNPST97"> <p><a class="existingWikiWord" href="/nlab/show/Igor+Bandos">Igor Bandos</a>, Kurt Lechner, Alexei Nurmagambetov, Paolo Pasti, <a class="existingWikiWord" href="/nlab/show/Dmitri+Sorokin">Dmitri Sorokin</a>, Mario Tonin, <em>Covariant Action for the Super-Five-Brane of M-Theory</em> (<a href="http://arxiv.org/abs/hep-th/9701149">arXiv:hep-th/9701149</a>)</p> </li> <li id="APPS97"> <p>Mina Aganagic, Jaemo Park, Costin Popescu, <a class="existingWikiWord" href="/nlab/show/John+Schwarz">John Schwarz</a>, <em>World-Volume Action of the M Theory Five-Brane</em> (<a href="http://arxiv.org/abs/hep-th/9701166">arXiv:hep-th/9701166</a>)</p> </li> </ul> <p>The 7-cocycle on the <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a> which gives the <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+6-algebra">supergravity Lie 6-algebra</a> appears in these articles (somewhat secretly) in equation (<a href="#BLNPST97">BLNPST, equation (9)</a>).</p> <p>See also</p> <ul> <li id="BPST"><a class="existingWikiWord" href="/nlab/show/Igor+Bandos">Igor Bandos</a>, Paolo Pasti, <a class="existingWikiWord" href="/nlab/show/Dmitri+Sorokin">Dmitri Sorokin</a>, Mario Tonin, <em>Superbrane Actions and Geometrical Approach</em> (<a href="http://arxiv.org/abs/hep-th/9705064">arXiv:hep-th/9705064</a>)</li> </ul> <p>The 7-cocycle for the M5-brane on the <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a> is equation (8.8) there.</p> <p>The interpretation of the cocycles for the <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a> and for the <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> as cocycles on “<a class="existingWikiWord" href="/nlab/show/extended+super-Minkowski+spacetime">extended super-Minkowski spacetime</a>” is due to</p> <ul> <li id="CAIB99"> <p>C. Chrysso‌malakos, <a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+de+Azc%C3%A1rraga">José de Azcárraga</a>, <a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+M.+Izquierdo">José M. Izquierdo</a>, C. Pérez Bueno, <em>The geometry of branes and extended superspaces</em>, Nucl. Phys. B <strong>567</strong> (2000) 293-330 [<a href="http://arxiv.org/abs/hep-th/9904137">arXiv:hep-th/9904137</a>, <a href="https://doi.org/10.1016/S0550-3213(99)00512-X">doi:10.1016/S0550-3213(99)00512-X</a>]</p> </li> <li id="Sakaguchi00"> <p>Makoto Sakaguchi: <em>IIB-Branes and New Spacetime Superalgebras</em>, JHEP 0004 (2000) 019 [<a href="https://arxiv.org/abs/hep-th/9909143">arXiv:hep-th/9909143</a>, <a href="https://iopscience.iop.org/article/10.1088/1126-6708/2000/04/019">doi:10.1088/1126-6708/2000/04/019</a>]</p> </li> <li id="AzcarragaIzquierdo01"> <p><a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+de+Azc%C3%A1rraga">José de Azcárraga</a>, <a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+Izquierdo">J. M. Izquierdo</a>, <em>Superalgebra cohomology, the geometry of extended superspaces and superbranes</em> (<a href="https://arxiv.org/abs/hep-th/0105125">arXiv:hep-th/0105125</a>)</p> </li> </ul> <p>See also <em><a class="existingWikiWord" href="/nlab/show/division+algebras+and+supersymmetry">division algebras and supersymmetry</a></em>.</p> <p>A corresponding refinement of the brane scan to a “brane bouquet” of <a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebra">super L-∞ algebra</a> <a class="existingWikiWord" href="/nlab/show/infinity-Lie+algebra+cohomology">extensions</a> (hence in <a class="existingWikiWord" href="/nlab/show/infinity-Lie+theory">infinity-Lie theory</a> via <a class="existingWikiWord" href="/schreiber/show/%E2%88%9E-Wess-Zumino-Witten+theory">∞-Wess-Zumino-Witten theory</a>) is discussed in</p> <ul> <li id="FiorenzaSatiSchreiber13"> <p><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/The+brane+bouquet">Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields</a></em>, International Journal of Geometric Methods in Modern Physics <strong>12</strong>:02 (2015) 1550018 (<a href="http://arxiv.org/abs/1308.5264">arXiv:1308.5264</a>)</p> </li> <li id="FiorenzaSatiSchreiber15"> <p><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a> <em><a class="existingWikiWord" href="/schreiber/show/The+WZW+term+of+the+M5-brane+and+differential+cohomotopy">The WZW term of the M5-brane and differential cohomotopy</a></em>, J. Math. Phys. 56, 102301 (2015) (<a href="http://arxiv.org/abs/1506.07557">arXiv:1506.07557</a>)</p> </li> <li id="FiorenzaSatiSchreiber16"> <p><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Rational+sphere+valued+supercocycles+in+M-theory+and+type+IIA+string+theory">Rational sphere valued supercocycles in M-theory and type IIA string theory</a></em> (<a href="http://arxiv.org/abs/1606.03206">arXiv:1606.03206</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Vincent+Braunack-Mayer">Vincent Braunack-Mayer</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Gauge+enhancement+of+Super+M-Branes">Gauge enhancement of Super M-Branes</a></em> (<a href="https://arxiv.org/abs/1806.01115">arXiv:1806.01115</a>)</p> </li> </ul> <p>These cohomological arguments also appear in what is called the “ectoplasm” method for invariants in <a class="existingWikiWord" href="/nlab/show/super+Yang-Mills+theory">super Yang-Mills theory</a> in</p> <ul> <li id="BossardHoweStelle09"> <p>G. Bossard, <a class="existingWikiWord" href="/nlab/show/Paul+Howe">Paul Howe</a>, K.S. Stelle, <em>A note on the UV behaviour of maximally supersymmetric Yang-Mills theories</em>, Phys. Lett. B682:137-142 (2009) (<a href="http://arxiv.org/abs/0908.3883">arXiv:0908.3883</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Paul+Howe">Paul Howe</a>, T. G. Pugh, K. S. Stelle, C. Strickland-Constable, <em>Ectoplasm with an Edge</em>, JHEP 1108:081,2011 (<a href="http://arxiv.org/abs/1104.4387">arXiv:1104.4387</a>)</p> </li> <li> <p>G. Bossard, <a class="existingWikiWord" href="/nlab/show/Paul+Howe">Paul Howe</a>, <a class="existingWikiWord" href="/nlab/show/Ulf+Lindstr%C3%B6m">Ulf Lindström</a>, <a class="existingWikiWord" href="/nlab/show/Kellogg+Stelle">Kellogg Stelle</a>, L. Wulff, <em>Integral invariants in maximally supersymmetric Yang-Mills theories</em> (<a href="http://arxiv.org/abs/1012.3142">arXiv:1012.3142</a>)</p> </li> </ul> <p>The connection is made in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Paul+Howe">Paul Howe</a>, O. Raetzel, <a class="existingWikiWord" href="/nlab/show/Ergin+Sezgin">Ergin Sezgin</a>, <em>On Brane Actions and Superembeddings</em>, JHEP 9808 (1998) 011 (<a href="http://arxiv.org/abs/hep-th/9804051">arXiv:hep-th/9804051</a>)</li> </ul> <p>The other brane scan, listing consistent asymptotic <a class="existingWikiWord" href="/nlab/show/AdS%2FCFT">AdS/CFT</a> <a class="existingWikiWord" href="/nlab/show/spacetimes">spacetimes</a>/<a class="existingWikiWord" href="/nlab/show/superconformal+algebras">superconformal algebras</a> is due to:</p> <ul> <li id="BlencoweDuff88"><a class="existingWikiWord" href="/nlab/show/Miles+P.+Blencowe">Miles P. Blencowe</a>, <a class="existingWikiWord" href="/nlab/show/Mike+Duff">Mike Duff</a>, <em>Supersingletons</em>, Physics letters B, <strong>203</strong> 3(1988) 229-236 (<a href="http://cds.cern.ch/record/184143">cds:184143</a>, <a href="http://dx.doi.org/10.1016/0370-2693(88)90544-8">doi:10.1016/0370-2693(88)90544-8</a>)</li> </ul> <p>with further developments discussed in</p> <ul> <li id="Duff09"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Duff">Michael Duff</a>, <em>Near-horizon brane-scan revived</em>, Nucl. Phys. B 810:193-209, 2009 (<a href="http://arxiv.org/abs/0804.3675">arXiv:0804.3675</a>)</p> </li> <li id="Duff21"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Duff">Michael Duff</a>, <em>The conformal brane-scan: an update</em> (<a href="https://arxiv.org/abs/2112.13784">arXiv:2112.13784</a>)</p> </li> </ul> <h3 id="ReferencesAdSBackgrounds">Anti de Sitter backgrounds</h3> <p>Discussion of Green-Schwarz strings on <a class="existingWikiWord" href="/nlab/show/super+anti+de+Sitter+spacetimes">super anti de Sitter spacetimes</a> includes the following.</p> <p>General:</p> <ul> <li id="HatsudaSakaguchi02"> <p>Machiko Hatsuda, Makoto Sakaguchi, <em>Wess-Zumino term for AdS superstring</em>, Phys.Rev. D66 (2002) 045020 (<a href="http://arxiv.org/abs/hep-th/0205092">arXiv:hep-th/0205092</a>, <a href="https://doi.org/10.1103/PhysRevD.66.045020">doi:10.1103/PhysRevD.66.045020</a>)</p> </li> <li id="HatsudaSakaguchi01"> <p>Machiko Hatsuda, Makoto Sakaguchi, <em>Wess-Zumino term for the AdS superstring and generalized Inonu-Wigner contraction</em>, Prog.Theor.Phys. 109 (2003) 853-867 (<a href="http://arxiv.org/abs/hep-th/0106114">arXiv:hep-th/0106114</a>)</p> </li> </ul> <h4 id="ReferencesAdS5Background"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>AdS</mi> <mn>5</mn></msub></mrow><annotation encoding="application/x-tex">AdS_5</annotation></semantics></math></h4> <p>The super 3-cocycle for the Green-Schwarz <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a> on the <a class="existingWikiWord" href="/nlab/show/super+anti+de+Sitter+spacetime">super anti de Sitter spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>AdS</mi> <mn>5</mn></msub><mo>×</mo><msup><mi>S</mi> <mn>5</mn></msup></mrow><annotation encoding="application/x-tex">AdS_5 \times S^5</annotation></semantics></math> (i.e. on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo stretchy="false">|</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mn>4</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>×</mo><mi>SO</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{SU(2,2 \vert 5)}{Spin(4,1)\times SO(5)}</annotation></semantics></math>) is originally due to</p> <ul> <li id="MetsaevTseytlin98"><a class="existingWikiWord" href="/nlab/show/Ruslan+Metsaev">Ruslan Metsaev</a>, <a class="existingWikiWord" href="/nlab/show/Arkady+Tseytlin">Arkady Tseytlin</a>, <em>Type IIB superstring action in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>AdS</mi> <mn>5</mn></msub><mo>×</mo><msup><mi>S</mi> <mn>5</mn></msup></mrow><annotation encoding="application/x-tex">AdS_5 \times S^5</annotation></semantics></math> background</em>, Nucl. Phys. <strong>B533</strong>:109-126, 1998 (<a href="https://arxiv.org/abs/hep-th/9805028">arXiv:hep-th/9805028</a>)</li> </ul> <p>However, a supersymmetric trivialization of this cocycle seems to have been obtained in</p> <ul> <li id="RoibanSiegel00">Radu Roiban, <a class="existingWikiWord" href="/nlab/show/Warren+Siegel">Warren Siegel</a>, <em>Superstrings on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>AdS</mi> <mn>5</mn></msub><mo>×</mo><msup><mi>S</mi> <mn>5</mn></msup></mrow><annotation encoding="application/x-tex">AdS_5 \times S^5</annotation></semantics></math> supertwistor space</em>, JHEP 0011:024, 2000 (<a href="https://arxiv.org/abs/hep-th/0010104">arXiv:hep-th/0010104</a>)</li> </ul> <p>see <a href="#HatsudaSakaguchi02">Hatsuda-Sakaguchi 02, around (1.2) and (2.6)</a>, <a href="#HatsudaSakaguchi01">Hatsuda-Sakaguchi 01, around (1.2)</a></p> <p>(amplified in <a href="https://arxiv.org/abs/1808.04470">arxiv:1808.04470, p. 5 and equation (5.5)</a>).</p> <p>See also</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Gleb+Arutyunov">Gleb Arutyunov</a>, <a class="existingWikiWord" href="/nlab/show/Sergey+Frolov">Sergey Frolov</a>, <em>Foundations of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>AdS</mi> <mn>5</mn></msub><mi>x</mi><msup><mi>S</mi> <mn>5</mn></msup></mrow><annotation encoding="application/x-tex">AdS_5 x S^5</annotation></semantics></math> Superstring. Part I</em>, J. Phys. A 42 (2009) (<a href="http://arxiv.org/abs/0901.4937">arXiv:0901.4937</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/John+Schwarz">John Schwarz</a>, <em>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>AdS</mi> <mn>5</mn></msub><mo>×</mo><msup><mi>S</mi> <mn>5</mn></msup></mrow><annotation encoding="application/x-tex">AdS_5 \times S^5</annotation></semantics></math> Superstring</em> (<a href="https://arxiv.org/abs/2004.09661">arXiv:2004.09661</a>)</p> </li> </ul> <h4 id="ReferencesAdS4Background"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>AdS</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">AdS_4</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>AdS</mi> <mn>7</mn></msub></mrow><annotation encoding="application/x-tex">AdS_7</annotation></semantics></math></h4> <p>For the <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Gleb+Arutyunov">Gleb Arutyunov</a>, <a class="existingWikiWord" href="/nlab/show/Sergey+Frolov">Sergey Frolov</a>, <em>Superstrings on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>AdS</mi> <mn>4</mn></msub><mo>×</mo><mi>ℂ</mi><msup><mi>P</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">AdS_4 \times \mathbb{C}P^3</annotation></semantics></math> as a Coset Sigma-model</em>, JHEP 0809:129,2008 (<a href="http://arxiv.org/abs/0806.4940">arXiv:0806.4940</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pietro+Fr%C3%A9">Pietro Fré</a>, <a class="existingWikiWord" href="/nlab/show/Pietro+Antonio+Grassi">Pietro Antonio Grassi</a>, <em>Pure Spinor Formalism for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Osp</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">|</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Osp(N|4)</annotation></semantics></math> backgrounds</em> (<a href="http://arxiv.org/abs/0807.0044">arXiv:0807.0044</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Riccardo+D%27Auria">Riccardo D'Auria</a>, <a class="existingWikiWord" href="/nlab/show/Pietro+Fr%C3%A9">Pietro Fré</a>, Pietro Antonio Grassi, Mario Trigiante, <em>Superstrings on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>AdS</mi> <mn>4</mn></msub><mo>×</mo><mi>ℂ</mi><msup><mi>P</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">AdS_4 \times \mathbb{C}P^3</annotation></semantics></math> from Supergravity</em> (<a href="http://arxiv.org/abs/0808.1282">arXiv:0808.1282</a>)</p> </li> <li> <p>D. V. Uvarov, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>AdS</mi> <mn>4</mn></msub><mi>x</mi><mi>ℂ</mi><msup><mi>P</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">AdS_4 x \mathbb{C}P^3</annotation></semantics></math> superstring in the light-cone gauge</em>, Nucl.Phys.B826:294-312,2010 (<a href="http://arxiv.org/abs/0906.4699">arXiv:0906.4699</a>)</p> </li> </ul> <p>For the <a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a>:</p> <ul> <li id="deWitPeetersPlefkaSevrin98"> <p><a class="existingWikiWord" href="/nlab/show/Bernard+de+Wit">Bernard de Wit</a>, <a class="existingWikiWord" href="/nlab/show/Kasper+Peeters">Kasper Peeters</a>, <a class="existingWikiWord" href="/nlab/show/Jan+Plefka">Jan Plefka</a>, Alexander Sevrin, <em>The M-Theory Two-Brane in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>AdS</mi> <mn>4</mn></msub><mo>×</mo><msup><mi>S</mi> <mn>7</mn></msup></mrow><annotation encoding="application/x-tex">AdS_4 \times S^7</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>AdS</mi> <mn>7</mn></msub><mo>×</mo><msup><mi>S</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">AdS_7 \times S^4</annotation></semantics></math></em>, Phys.Lett. B443 (1998) 153-158 (<a href="http://arxiv.org/abs/hep-th/9808052">arXiv:hep-th/9808052</a>)</p> </li> <li> <p>Makoto Sakaguchi, Hyeonjoon Shin, Kentaroh Yoshida, <em>Semiclassical Analysis of M2-brane in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>AdS</mi> <mn>4</mn></msub><mo>×</mo><msup><mi>S</mi> <mn>7</mn></msup><mo stretchy="false">/</mo><msub><mi>ℤ</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">AdS_4 \times S^7 / \mathbb{Z}_k</annotation></semantics></math></em>, JHEP 1012:012,2010 (<a href="http://arxiv.org/abs/1007.3354">arXiv:1007.3354</a>)</p> </li> </ul> <p>For the <a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a> and the <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a>:</p> <ul> <li id="Claus98"> <p><a class="existingWikiWord" href="/nlab/show/Piet+Claus">Piet Claus</a>, <em>Super M-brane actions in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>adS</mi> <mn>4</mn></msub><mo>×</mo><msup><mi>S</mi> <mn>7</mn></msup></mrow><annotation encoding="application/x-tex">adS_4 \times S^7</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>adS</mi> <mn>7</mn></msub><mo>×</mo><msup><mi>S</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">adS_7 \times S^4</annotation></semantics></math></em>, Phys. Rev. D <strong>59</strong> (1999) 066003 [<a href="http://arxiv.org/abs/hep-th/9809045">arXiv:hep-th/9809045</a>, <a href="https://doi.org/10.1103/PhysRevD.59.066003">doi:10.1103/PhysRevD.59.066003</a>]</p> </li> <li> <p>Makoto Sakaguchi, Kentaroh Yoshida, <em>Open M-branes on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>AdS</mi> <mrow><mn>4</mn><mo stretchy="false">/</mo><mn>7</mn></mrow></msub><mo>×</mo><msup><mi>S</mi> <mrow><mn>7</mn><mo stretchy="false">/</mo><mn>4</mn></mrow></msup></mrow><annotation encoding="application/x-tex">AdS_{4/7} \times S^{7/4}</annotation></semantics></math> Revisited</em>, Nucl.Phys. B714 (2005) 51-66 (<a href="http://arxiv.org/abs/hep-th/0405109">arXiv:hep-th/0405109</a>)</p> </li> </ul> <h3 id="selfdual_strings_in_6d">Self-dual strings in 6d</h3> <p>Discussion of the <a class="existingWikiWord" href="/nlab/show/self-dual+string">self-dual string</a> in 6d as a Green-Schwarz-type sigma model includes</p> <ul> <li>Par Arvidsson, Erik Flink, <a class="existingWikiWord" href="/nlab/show/Mans+Henningson">Mans Henningson</a>, <em>Supersymmetric coupling of a self-dual string to a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2,0)</annotation></semantics></math> tensor multiplet background</em>, JHEP0311:015,2003 (<a href="http://arxiv.org/abs/hep-th/0309244">arXiv:hep-th/0309244</a>)</li> </ul> <h3 id="ReferencesSupergravityBackgroundEquationsOfMotion">General curved backgrounds and Supergravity background equations of motion</h3> <p>The consistency of the Green-Schwarz action functional for the superstring in a <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> <a class="existingWikiWord" href="/nlab/show/background+gauge+field">background</a> should be equivalent to the background satiyfying the <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> <a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a>. For the superstring this is due to</p> <ul> <li id="BergshoeffSezginTownsend86"><a class="existingWikiWord" href="/nlab/show/Eric+Bergshoeff">Eric Bergshoeff</a>, <a class="existingWikiWord" href="/nlab/show/Ergin+Sezgin">Ergin Sezgin</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Townsend">Paul Townsend</a>, <em>Superstring actions in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>10</mn></mrow><annotation encoding="application/x-tex">D = 3, 4, 6, 10</annotation></semantics></math> curved superspace</em>, Phys.Lett., B169, 191, (1986) (<a href="http://inspirehep.net/record/223138/?ln=en">spire</a>)</li> </ul> <p>and for the supermembrane due to</p> <ul> <li id="BergshoeffSezginTownsend87"><a class="existingWikiWord" href="/nlab/show/Eric+Bergshoeff">Eric Bergshoeff</a>, <a class="existingWikiWord" href="/nlab/show/Ergin+Sezgin">Ergin Sezgin</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Townsend">Paul Townsend</a>, <em>Supermembranes and eleven dimensional supergravity</em>, Phys.Lett. B189 (1987) 75-78, In <a class="existingWikiWord" href="/nlab/show/Mike+Duff">Mike Duff</a>, (ed.), <em><a class="existingWikiWord" href="/nlab/show/The+World+in+Eleven+Dimensions">The World in Eleven Dimensions</a></em> 69-72 (<a href="http://inspirehep.net/record/248230?ln=en">spire</a>)</li> </ul> <p>with a quick re-derivation using that the <a class="existingWikiWord" href="/nlab/show/torsion+constraints+in+supergravity">torsion constraint in 11d supergravity</a> already imples the sugra equations of motion is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Paul+Howe">Paul Howe</a>, <a class="existingWikiWord" href="/nlab/show/Ergin+Sezgin">Ergin Sezgin</a>, section 2.1 of <em>The supermembrane revisited</em>, (<a href="http://arxiv.org/abs/hep-th/0412245">arXiv:hep-th/0412245</a>)</li> </ul> <p>These authors amplify the role of closed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(p+2)</annotation></semantics></math>-forms in super <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-brane backgrounds (p. 3) and clearly state the consistency conditions for the <a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a> in a curved backround in terms of the <a class="existingWikiWord" href="/nlab/show/Bianchi+identities">Bianchi identities</a> on p. 7-8, amounting to the statment that the 4-form field strength has to be the pullback of the cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>ψ</mi><mo>¯</mo></mover><mo>∧</mo><msup><mi>e</mi> <mi>a</mi></msup><mo>∧</mo><msup><mi>e</mi> <mi>b</mi></msup><mo>∧</mo><msup><mi>Γ</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msup><mi>ψ</mi></mrow><annotation encoding="application/x-tex">\overline{\psi}\wedge e^a \wedge e^b \wedge \Gamma^{a b} \psi</annotation></semantics></math> plus the <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a> <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> and has to be closed.</p> <p>That the <a class="existingWikiWord" href="/nlab/show/heterotic+supergravity">heterotic supergravity</a> equations of motion are sufficient for the 3-form super field strength <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> to be closed was first argued in</p> <ul> <li id="Nilsson81"><a class="existingWikiWord" href="/nlab/show/Bengt+Nilsson">Bengt Nilsson</a>, <em>Simple 10-dimensional supergravity in superspace</em>, Nuclear Physics B188 (1981) 176-192 (<a href="http://inspirehep.net/record/164253?ln=de">spire</a>)</li> </ul> <p>and the computation there was highlighted and a little simplified around p. 17 of</p> <ul> <li id="Witten86"><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>Twistor-like transform in ten dimensions</em>, Nuclear Physics B266 (1986) (<a href="http://inspirehep.net/record/214192/?ln=en">spire</a>)</li> </ul> <p>A more comprehensive result arguing that the heterotic supergravity equations of motion of the background are not just sufficient but also necessary for (and hence equivalent to) the heterotic GS-string on that background being consistent was then claimed in</p> <ul> <li id="ShapiroTaylor87"><a class="existingWikiWord" href="/nlab/show/Joel+Shapiro">Joel Shapiro</a>, Cyrus Taylor, <em>Superspace supergravity from the superstring</em>, Physics letter B <strong>186</strong> 1 (1987) [<a href="https://doi.org/10.1016/0370-2693(87)90514-4">doi:10.1016/0370-2693(87)90514-4</a>]</li> </ul> <p>Discussion of this with the heterotic gauge-field included (hence including the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+anomaly+cancellation">Green-Schwarz anomaly cancellation</a>) is in</p> <ul> <li id="AtickDharRatra86"><a class="existingWikiWord" href="/nlab/show/Joseph+Atick">Joseph Atick</a>, Avinash Dhar, Bharat Ratra, <em>Superspace Formulation of Ten-dimensional <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">N=1</annotation></semantics></math> Supergravity Coupled to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">N=1</annotation></semantics></math> Super Yang-Mills Theory</em>, Phys.Rev. D33 (1986) 2824 (<a href="https://inspirehep.net/record/219734">spire</a>, <a href="http://ccdb5fs.kek.jp/cgi-bin/img/allpdf?198602074">pdf</a>)</li> </ul> <p>Similar arguments for the <a class="existingWikiWord" href="/nlab/show/type+II+string">type II string</a> in <a class="existingWikiWord" href="/nlab/show/type+II+supergravity">type II supergravity</a> appeared in</p> <ul> <li id="GHMNT85"><a class="existingWikiWord" href="/nlab/show/Marcus+Grisaru">Marcus Grisaru</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Howe">Paul Howe</a>, L. Mezincescu, <a class="existingWikiWord" href="/nlab/show/Bengt+Nilsson">Bengt Nilsson</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Townsend">Paul Townsend</a>, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">N=2</annotation></semantics></math>-Superstring in a supergravity background</em>, Physics Letters Volume 162B, number 1,2,3 (1985) (<a href="http://inspirehep.net/record/17010">spire</a>)</li> </ul> <p>and for GS sigma-model <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a> in</p> <ul> <li id="CGNSW97"><a class="existingWikiWord" href="/nlab/show/Martin+Cederwall">Martin Cederwall</a>, Alexander von Gussich, <a class="existingWikiWord" href="/nlab/show/Bengt+Nilsson">Bengt Nilsson</a>, Per Sundell, Anders Westerberg, <em>The Dirichlet Super-p-Branes in Ten-Dimensional Type IIA and IIB Supergravity</em>, Nucl.Phys. B490 (1997) 179-201 (<a href="http://arxiv.org/abs/hep-th/9611159">arXiv:hep-th/9611159</a>)</li> </ul> <p>That the <a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a> <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> is consistent on backgrounds of <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a> that satisfy their equations of motion is discussed in (<a href="#BergshoeffSezginTownsend87">Bergshoeff-Sezgin-Townsend 87</a>).</p> <p>The role of the 4-form here is also amplified around (2.29) in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Igor+Bandos">Igor Bandos</a>, Carlos Meliveo, <em>Supermembrane interaction with dynamical D=4 N=1 supergravity. Superfield Lagrangian description and spacetime equations of motion</em> (<a href="https://arxiv.org/abs/arXiv:1205.5885">arXiv:1205.5885</a>)</li> </ul> <p>and in section 2.2 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Igor+Bandos">Igor Bandos</a>, Carlos Meliveo, <em>Three form potential in (special) minimal supergravity superspace and supermembrane supercurrent</em> (<a href="http://arxiv.org/abs/1107.3232">arXiv:1107.3232</a>)</li> </ul> <p>following</p> <ul> <li id="OvrutWaldram97"><a class="existingWikiWord" href="/nlab/show/Burt+Ovrut">Burt Ovrut</a>, <a class="existingWikiWord" href="/nlab/show/Daniel+Waldram">Daniel Waldram</a>, <em>Membranes and Three-form Supergravity</em>, Nucl.Phys. B506 (1997) 236-266 (<a href="http://arxiv.org/abs/hep-th/9704045">arXiv:hep-th/9704045</a>)</li> </ul> <p>See also</p> <ul> <li>Bernard de Wit, Kasper Peeters, Jan Plefka, <em>Superspace Geometry for Supermembrane Backgrounds</em>, Nucl.Phys. B532 (1998) 99-123 (<a href="http://arxiv.org/abs/hep-th/9803209">arXiv:hep-th/9803209</a>)</li> </ul> <p>All this is actually subsumed by imposing the <a class="existingWikiWord" href="/nlab/show/Bianchi+identities">Bianchi identities</a> of the corresponding <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a> etc. in “rheonomic parameterization”, of the <em><a class="existingWikiWord" href="/nlab/show/D%27Auria-Fr%C3%A9+formulation+of+supergravity">D'Auria-Fré formulation of supergravity</a></em>, this is discussed in (<a href="#AFFFTT98">AFFFTT 98, section 3.1</a>, <a href="#FreGrassi07">Fré-Grassi 07</a>).</p> <p>Discussion including also the <a class="existingWikiWord" href="/nlab/show/RR-field">RR-field</a> background includes</p> <ul> <li>R. R. Metsaev, <em>Type IIB Green-Schwarz superstring in plane wave Ramond-Ramond background</em> (<a href="http://arxiv.org/abs/hep-th/0112044">arXiv:hep-th/0112044</a>)</li> </ul> <div> <h3 id="ReferencesMicroscopicAdSCFTViapBraneSigmaModesl">Microscopic AdS/CFT via <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-brane sigma-models</h3> <p>Over a decade before the <a href="AdS-CFT+correspondence#ReferencesOriginModernAdSCFTDUality">modern formulation</a> of the <a class="existingWikiWord" href="/nlab/show/AdS-CFT+correspondence">AdS-CFT correspondence</a>, a candicate “microscopic” explanation was observed:</p> <div class="float_right_image" style="margin: -20px 0px 0px 15px"> <img src="/nlab/files/pBraneNearBlackBrane-240624.jpg" width="400px" /> </div> <p><a class="existingWikiWord" href="/nlab/show/immersion">Immersing</a> the <a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a> of a <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> <a class="existingWikiWord" href="/nlab/show/super+p-brane">super <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>p</mi> </mrow> <annotation encoding="application/x-tex">p</annotation> </semantics> </math>-brane</a> (hence a “light” <a class="existingWikiWord" href="/nlab/show/brane">brane</a> without <a class="existingWikiWord" href="/nlab/show/backreaction">backreaction</a>) along the <a class="existingWikiWord" href="/nlab/show/near+horizon+geometry">near horizon geometry</a> (an <a class="existingWikiWord" href="/nlab/show/anti+de+Sitter+spacetime">AdS</a> <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> solution) of its own <a class="existingWikiWord" href="/nlab/show/black+brane">black brane</a> incarnation (hence the “heavy” incarnation of the same brane, causing <a class="existingWikiWord" href="/nlab/show/backreaction">backreaction</a>), its <a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a> fluctuations (after <a class="existingWikiWord" href="/nlab/show/super-diffeomorphism">super-diffeomorphism</a> <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixing</a>) are described by the corresponding <a class="existingWikiWord" href="/nlab/show/superconformal+field+theory">superconformal field theory</a> (exhibited by <a class="existingWikiWord" href="/nlab/show/superconformal+multiplets">superconformal multiplets</a> such as “<em><a class="existingWikiWord" href="/nlab/show/supersingletons">supersingletons</a></em>”).</p> <p>The original observation for the <a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a>:</p> <ul> <li id="BDPS87"> <p><a class="existingWikiWord" href="/nlab/show/Eric+Bergshoeff">Eric Bergshoeff</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Duff">Michael Duff</a>, <a class="existingWikiWord" href="/nlab/show/Christopher+Pope">Christopher Pope</a>, <a class="existingWikiWord" href="/nlab/show/Ergin+Sezgin">Ergin Sezgin</a>: <em>Supersymmetric Supermembrane Vacua and Singletons</em>, Phys. Lett. B <strong>199</strong> (1987) 69-74 [<a href="https://doi.org/10.1016/0370-2693(87)91465-1">doi:10.1016/0370-2693(87)91465-1</a>, <a href="https://inspirehep.net/literature/250244">InSpire:250244</a>]</p> </li> <li id="BlencoweDuff88"> <p><a class="existingWikiWord" href="/nlab/show/Miles+P.+Blencowe">Miles P. Blencowe</a>, <a class="existingWikiWord" href="/nlab/show/Mike+Duff">Mike Duff</a>, <em>Supersingletons</em>, Physics letters B, <strong>203</strong> 3 (1988) 229-236 [<a href="http://dx.doi.org/10.1016/0370-2693(88)90544-8">doi:10.1016/0370-2693(88)90544-8</a>, <a href="http://cds.cern.ch/record/184143">cds:184143</a>]</p> </li> <li id="DGGGTT98"> <p><a class="existingWikiWord" href="/nlab/show/Gianguido+Dall%27Agata">Gianguido Dall'Agata</a>, Davide Fabbri, Christophe Fraser, <a class="existingWikiWord" href="/nlab/show/Pietro+Fr%C3%A9">Pietro Fré</a>, Piet Termonia, <a class="existingWikiWord" href="/nlab/show/Mario+Trigiante">Mario Trigiante</a>, <em>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Osp</mi><mo stretchy="false">(</mo><mn>8</mn><mo stretchy="false">|</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Osp(8|4)</annotation></semantics></math> singleton action from the supermembrane</em>, Nucl. Phys. B <strong>542</strong> (1999) 157-194 [<a href="https://doi.org/10.1016/S0550-3213(98)00765-2">doi:10.1016/S0550-3213(98)00765-2</a>, <a href="http://arxiv.org/abs/hep-th/9807115">arXiv:hep-th/9807115</a>]</p> </li> </ul> <p>with popular exposition in:</p> <ul> <li id="DuffSutton88"><a class="existingWikiWord" href="/nlab/show/Mike+Duff">Mike Duff</a>, Christine Sutton: <em>The Membrane at the End of the Universe</em>, New Scientist <strong>118</strong> (1988) 67-71 [<a href="http://inspirehep.net/record/268230">inspire:268230</a>, ISSN:0028-6664]</li> </ul> <p>Further discussion including also <a class="existingWikiWord" href="/nlab/show/M5-branes">M5-branes</a> and <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a>:</p> <ul> <li id="ClausKalloshProeyen97"> <p><a class="existingWikiWord" href="/nlab/show/Piet+Claus">Piet Claus</a>, <a class="existingWikiWord" href="/nlab/show/Renata+Kallosh">Renata Kallosh</a>, <a class="existingWikiWord" href="/nlab/show/Antoine+Van+Proeyen">Antoine Van Proeyen</a>, <em>M 5-brane and superconformal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,2)</annotation></semantics></math> tensor multiplet in 6 dimensions</em>, Nucl. Phys. B <strong>518</strong> (1998) 117-150 [<a href="https://doi.org/10.1016/S0550-3213(98)00137-0">doi:10.1016/S0550-3213(98)00137-0</a>, <a href="http://arxiv.org/abs/hep-th/9711161">arXiv:hep-th/9711161</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Piet+Claus">Piet Claus</a>, <a class="existingWikiWord" href="/nlab/show/Renata+Kallosh">Renata Kallosh</a>, J. Kumar, <a class="existingWikiWord" href="/nlab/show/Paul+K.+Townsend">Paul K. Townsend</a>, <a class="existingWikiWord" href="/nlab/show/Antoine+Van+Proeyen">Antoine Van Proeyen</a>, <em>Conformal Theory of M2, D3, M5 and “D1+D5” Branes</em>, JHEP 9806 (1998) 004 [<a href="https://doi.org/10.1088/1126-6708/1998/06/004">doi:10.1088/1126-6708/1998/06/004</a>, <a href="https://arxiv.org/abs/hep-th/9801206">arXiv:hep-th/9801206</a>]</p> </li> <li id="PastiSorokinTonin99"> <p><a class="existingWikiWord" href="/nlab/show/Paolo+Pasti">Paolo Pasti</a>, <a class="existingWikiWord" href="/nlab/show/Dmitri+Sorokin">Dmitri Sorokin</a>, <a class="existingWikiWord" href="/nlab/show/Mario+Tonin">Mario Tonin</a>, <em>Branes in Super-AdS Backgrounds and Superconformal Theories</em>, Proceedings, <em>International Workshop on Supersymmetries and Quantum Symmetries (SQS’99)</em>, Moscow (July 27-31, 1999) [<a href="http://arxiv.org/abs/hep-th/9912076">arXiv:hep-th/9912076</a>, <a href="https://inspirehep.net/literature/511348">inspire:511348</a>]</p> </li> <li> <p>C. Grojean, J. Mourad, <em>Super fivebranes near the boundary of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>AdS</mi> <mn>7</mn></msub><mo>×</mo><msup><mi>S</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">AdS_7 \times S^4</annotation></semantics></math></em>, Nuclear Physics B <strong>567</strong> 1–2 (2000) 133-150 [<a href="https://doi.org/10.1016/S0550-3213(99)00335-1">doi:10.1016/S0550-3213(99)00335-1</a>, <a href="https://arxiv.org/abs/hep-th/9903164">arXiv:hep-th/9903164</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Alexei+Nurmagambetov">Alexei Nurmagambetov</a>, I. Y. Park <em>On the M5 and the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">AdS</mi> <mn>7</mn></msub><mo stretchy="false">/</mo><msub><mi>CFT</mi> <mn>6</mn></msub></mrow><annotation encoding="application/x-tex">\mathrm{AdS}_7/CFT_6</annotation></semantics></math> correspondence</em>, Physics Letters B</p> <p><strong>524</strong> 1–2 (2002) 185-191 [<a href="https://doi.org/10.1016/S0370-2693(01)01375-2">doi:10.1016/S0370-2693(01)01375-2</a>, <a href="https://arxiv.org/abs/hep-th/0110192">arXiv:hep-th/0110192</a>]</p> </li> </ul> <p>Review:</p> <ul> <li id="Duff98"> <p><a class="existingWikiWord" href="/nlab/show/Mike+Duff">Mike Duff</a>, <em>Anti-de Sitter space, branes, singletons, superconformal field theories and all that</em>, Int. J. Mod. Phys. A <strong>14</strong> (1999) 815-844 [<a href="https://doi.org/10.1142/S0217751X99000403">doi:10.1142/S0217751X99000403</a>, <a href="https://arxiv.org/abs/hep-th/9808100">arXiv:hep-th/9808100</a>]</p> </li> <li id="Duff99"> <p><a class="existingWikiWord" href="/nlab/show/Mike+Duff">Mike Duff</a>, <em>TASI Lectures on Branes, Black Holes and Anti-de Sitter Space</em>, lectures at <em><a href="https://inspirehep.net/conferences/972153">9th CRM Summer School: Theoretical Physics at the End of the 20th Century</a></em> at <a href="https://inspirehep.net/conferences/972127">TASI 99</a> (1999) [<a href="https://arxiv.org/abs/hep-th/9912164">arXiv:hep-th/9912164</a>, <a href="https://inspirehep.net/literature/511822">inspire:511822</a>]</p> </li> </ul> <p>The resulting <a class="existingWikiWord" href="/nlab/show/superconformal+algebra">super-conformal</a> <a class="existingWikiWord" href="/nlab/show/brane+scan">brane scan</a>:</p> <ul> <li id="Duff09"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Duff">Michael Duff</a>, <em>Near-horizon brane-scan revived</em>, Nucl. Phys. B <strong>810</strong> (2009) 193-209 [<a href="https://doi.org/10.1016/j.nuclphysb.2008.11.001">doi:10.1016/j.nuclphysb.2008.11.001</a>, <a href="http://arxiv.org/abs/0804.3675">arXiv:0804.3675</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael+Duff">Michael Duff</a>, <em>The conformal brane-scan: an update</em>, J. High Energ. Phys. <strong>2022</strong> 54 (2022) [<a href="https://doi.org/10.1007/JHEP06(2022)054">doi:10.1007/JHEP06(2022)054</a>, <a href="https://arxiv.org/abs/2112.13784">arXiv:2112.13784</a>]</p> </li> </ul> <p>Related:</p> <ul> <li>Hironori Mori, Satoshi Yamaguchi, §3 in: <em>M5-branes and Wilson Surfaces in AdS7/CFT6 Correspondence</em>, Phys. Rev. D <strong>90</strong> (2014) 026005 [<a href="https://doi.org/10.1103/PhysRevD.90.026005">doi:10.1103/PhysRevD.90.026005</a>, <a href="https://arxiv.org/abs/1404.0930">arXiv:1404.0930</a>]</li> </ul> <p>Analogous discussion for embeddings with less supersymmetry, corresponding to <a class="existingWikiWord" href="/nlab/show/defect+QFT">defects</a></p> <ul> <li> <p>Nadav Drukker, Simone Giombi, <a class="existingWikiWord" href="/nlab/show/Arkady+A.+Tseytlin">Arkady A. Tseytlin</a>, Xinan Zhou, <em>Defect CFT in the 6d <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2,0)</annotation></semantics></math> theory from M2 brane dynamics in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>AdS</mi> <mn>7</mn></msub><mo>×</mo><msup><mi>S</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">AdS_7 \times S^4</annotation></semantics></math></em>, J. High Energ. Phys. <strong>2020</strong> 101 (2020) [<a href="https://doi.org/10.1007/JHEP07(2020)101">doi:10.1007/JHEP07(2020)101</a>, <a href="https://arxiv.org/abs/2004.04562">arXiv:2004.04562</a>]</p> </li> <li> <p>Varun Gupta, <em>Holographic M5 branes in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">AdS</mi> <mn>7</mn></msub><mo>×</mo><msup><mi>S</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\mathrm{AdS}_7 \times S^4</annotation></semantics></math></em>, J. High Energ. Phys. <strong>2021</strong> 32 (2021) [<a href="https://doi.org/10.1007/JHEP12(2021)032">doi:10.1007/JHEP12(2021)032</a>, <a href="https://arxiv.org/abs/2109.08551">arXiv:2109.08551</a>]</p> </li> <li> <p>Varun Gupta, <em>More Holographic M5 branes in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi mathvariant="normal">AdS</mi> <mn>7</mn></msub><mo>×</mo><msup><mi>S</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\mathrm{AdS}_7 \times S^4</annotation></semantics></math></em>, Phys. Lett. B <strong>853</strong> (2024) 138650 [<a href="https://doi.org/10.1016/j.physletb.2024.138650">doi:10.1016/j.physletb.2024.138650</a>, <a href="https://arxiv.org/abs/2301.02528">arXiv:2301.02528</a>]</p> </li> </ul> </div> <h3 id="ReferencesConservedCurrentAlgebra">Conserved current algebra</h3> <p>That higher WZW functionals and hence Green-Schwarz super <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-brane action functionals have <a class="existingWikiWord" href="/nlab/show/conserved+current">conserved current</a> <a class="existingWikiWord" href="/nlab/show/BPS+charge">BPS charge</a> algebras which are <a href="super+Poincare+Lie+algebra#PolyvectorExtensions">polyvector extensions</a> of the supersymmetry algebras was observed in</p> <ul> <li id="AGIT89"><a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+de+Azc%C3%A1rraga">José de Azcárraga</a>, <a class="existingWikiWord" href="/nlab/show/Jerome+Gauntlett">Jerome Gauntlett</a>, <a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+Izquierdo">J.M. Izquierdo</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Townsend">Paul Townsend</a>, <em>Topological extensions of the supersymmetry algebra for extended objects</em>, Phys. Rev. Lett. 63 (1989) 2443 (<a href="http://inspirehep.net/record/26393?ln=en">spire</a>)</li> </ul> <p>reviewed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+de+Azc%C3%A1rraga">José de Azcárraga</a>, <a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+M.+Izquierdo">José M. Izquierdo</a>, section 8.8. of <em><a class="existingWikiWord" href="/nlab/show/Lie+Groups%2C+Lie+Algebras%2C+Cohomology+and+Some+Applications+in+Physics">Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics</a></em> , Cambridge monographs of mathematical physics, (1995)</li> </ul> <p>and generalized to super-<a class="existingWikiWord" href="/nlab/show/Lie+n-algebras">Lie n-algebras</a> of BPS charges in</p> <ul> <li id="SatiSchreiber15"><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Lie+n-algebras+of+BPS+charges">Lie n-algebras of BPS charges</a></em> (<a href="https://arxiv.org/abs/1507.08692">arXiv:1507.08692</a>)</li> </ul> <p>This is for branes in the old <a class="existingWikiWord" href="/nlab/show/brane+scan">brane scan</a> (<a class="existingWikiWord" href="/nlab/show/strings">strings</a>, <a class="existingWikiWord" href="/nlab/show/membranes">membranes</a>, <a class="existingWikiWord" href="/nlab/show/NS5-branes">NS5-branes</a>), excluding <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a> and <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a>.</p> <p>The generalization oft this perspective to the <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> is discussed in</p> <ul> <li id="SorokinTownsend97"><a class="existingWikiWord" href="/nlab/show/Dmitri+Sorokin">Dmitri Sorokin</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Townsend">Paul Townsend</a>, <em>M-theory superalgebra from the M-5-brane</em>, Phys.Lett. B412 (1997) 265-273 (<a href="http://arxiv.org/abs/hep-th/9708003">arXiv:hep-th/9708003</a>)</li> </ul> <p>and the generalizatin to <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a> is discussed in</p> <ul> <li>Hanno Hammer, <em>Topological Extensions of Noether Charge Algebras carried by D-p-branes</em>, Nucl.Phys. B521 (1998) 503-546 (<a href="http://arxiv.org/abs/hep-th/9711009">arXiv:hep-th/9711009</a>)</li> </ul> <h3 id="symmetry"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>-Symmetry</h3> <p>The existence of <a class="existingWikiWord" href="/nlab/show/kappa-symmetry">kappa-symmetry</a> was first noticed around</p> <ul> <li id="Siegel83"> <p>Warren Siegel, <em>Hidden Local Supersymmetry In The Supersymmetric Particle Action</em> Phys. Lett. B 128, 397 (1983)</p> </li> <li id="Siegel84"> <p>Warren Siegel, <em>Light Cone Analysis Of Covariant Superstring</em> , Nucl. Phys. B 236, 311 (1984).</p> </li> <li id="GreenSchwarz"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Green">Michael Green</a>, <a class="existingWikiWord" href="/nlab/show/John+Schwarz">John Schwarz</a>, <em>Covariant Description Of Superstrings</em> , Phys. Lett. B 136, 367 (1984) (<a href="http://adsabs.harvard.edu/abs/1984PhLB..136..367G">web</a>)</p> </li> </ul> <p>The meaning of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>-symmetry in terms of the <a class="existingWikiWord" href="/nlab/show/supergeometry">super</a>-<a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a> of super-<a class="existingWikiWord" href="/nlab/show/target+space">target space</a> is discussed in</p> <ul> <li id="McArthur"><a class="existingWikiWord" href="/nlab/show/Ian+N.+McArthur">Ian N. McArthur</a>, <em>Kappa-Symmetry of Green-Schwarz Actions in Coset Superspaces</em>, Nucl. Phys. B <strong>573</strong> (2000) 811-829 [<a href="https://doi.org/10.1016/S0550-3213(99)00800-7">https://doi.org/10.1016/S0550-3213(99)00800-7</a>]</li> </ul> <p>[<a href="http://arxiv.org/abs/hep-th/9908045">arXiv:hep-th/9908045</a>]</p> <ul> <li id="GKW"><a class="existingWikiWord" href="/nlab/show/Joaquim+Gomis">Joaquim Gomis</a>, Kiyoshi Kamimura, <a class="existingWikiWord" href="/nlab/show/Peter+West">Peter West</a>, <em>Diffeomorphism, kappa transformations and the theory of non-linear realisations</em> (<a href="http://arxiv.org/abs/hep-th/0607104">arXiv:hep-th/0607104</a>)</li> </ul> <p>Discussion from the point of view of <a class="existingWikiWord" href="/nlab/show/D%27Auria-Fr%C3%A9+formulation+of+supergravity">D'Auria-Fré formulation of supergravity</a> is in</p> <ul> <li id="AFFFTT98"> <p>Gianguido Dall’Agata, Davide Fabbri, Christophe Fraser, <a class="existingWikiWord" href="/nlab/show/Pietro+Fr%C3%A9">Pietro Fré</a>, Piet Termonia, Mario Trigiante, <em>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Osp</mi><mo stretchy="false">(</mo><mn>8</mn><mo stretchy="false">|</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Osp(8|4)</annotation></semantics></math> singleton action from the supermembrane</em>, Nucl. Phys. B542:157-194,1999, (<a href="http://arxiv.org/abs/hep-th/9807115">arXiv:hep-th/9807115</a>)</p> </li> <li id="FreGrassi07"> <p><a class="existingWikiWord" href="/nlab/show/Pietro+Fr%C3%A9">Pietro Fré</a>, <a class="existingWikiWord" href="/nlab/show/Pietro+Antonio+Grassi">Pietro Antonio Grassi</a>, <em>Pure Spinors, Free Differential Algebras, and the Supermembrane</em>, Nucl. Phys. B763:1-34,2007 (<a href="http://arxiv.org/abs/hep-th/0606171">arXiv:hep-th/0606171</a>)</p> </li> </ul> <h3 id="open_branes_ending_on_other_branes">Open branes ending on other branes</h3> <p>Discussion of the Green-Schwarz action for the open <a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a> ending on the <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> is in</p> <ul> <li> <p>C. S. Chu, <a class="existingWikiWord" href="/nlab/show/Ergin+Sezgin">Ergin Sezgin</a>, <em>M-Fivebrane from the Open Supermembrane</em>, JHEP 9712 (1997) 001 (<a href="http://arxiv.org/abs/hep-th/9710223">arXiv:hep-th/9710223</a>)</p> </li> <li> <p>Ph. Brax, J. Mourad, <em>Open Supermembranes Coupled to M-Theory Five-Branes</em>, Phys. Lett. B416 (1998) 295-302 (<a href="http://arxiv.org/abs/hep-th/9707246">arXiv:hep-th/9707246</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on November 10, 2024 at 08:56:08. 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